Normal distribution

Template:Short description Template:Redirect Template:More citations needed Template:Use mdy dates {{#invoke:infobox3cols|infoboxTemplate | bodyclass = ib-prob-dist | templatestyles = Infobox probability distribution/styles.css | bodystyle = {{#if:|width: {{{box_width}}};}}

| title = Normal distribution | subheader =

| image1 = {{#if: File:Normal Distribution PDF.svg|

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| caption1 = The red curve is the standard normal distribution.

| image2 = {{#if: File:Normal Distribution CDF.svg|

Cumulative distribution function

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| caption2 =

| label1 = Notation | data1 = <math>\mathcal{N}(\mu,\sigma^2)</math>

| label2 = Parameters | data2 = {{#if: | | <math>\mu\in\R</math> = mean (location)
<math>\sigma^2\in\R_{>0}</math> = variance (squared scale) }} | data2a = <math>\mu\in\R</math> = mean (location)
<math>\sigma^2\in\R_{>0}</math> = variance (squared scale) | data2b =

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| label4 = {{#switch: density

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| data4 = {{#if: | | <math>\frac{1}{\sqrt{2\pi\sigma^2 }} | data4a = <math>\frac{1}{\sqrt{2\pi\sigma^2 | data4b =

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| label6 = Quantile | data6 =

| label7 = Mean | data7 = {{#if: | | }} | data7a = | data7b =

| label8 = Median | data8 = {{#if: | | }} | data8a = | data8b =

| label9 = Mode | data9 = {{#if: | | }} | data9a = | data9b =

| label10 = {{#switch: density

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| label11 = MAD | data11 = {{#if: | | }} | data11a = | data11b =

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| label14 = Excess kurtosis | data14 = {{#if: | | }} | data14a = | data14b =

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| label16 = MGF | data16 = {{#if: | | }} | data16a = | data16b =

| label17 = CF | data17 = {{#if: | | }} | data17a = | data17b =

| label18 = PGF | data18 = {{#if: | | }} | data18a = | data18b =

| label19 = Fisher information | data19 = {{#if: | | }} | data19a = | data19b =

| label20 = Kullback–Leibler divergence | data20 =

| label21 = Jensen-Shannon divergence | data21 =

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| label23 = Expected shortfall | data23 =

}}{{#invoke:Check for unknown parameters|check|unknown=Template:Main other|preview=Page using Template:Infobox probability distribution with unknown parameter "_VALUE_"|ignoreblank=y| box_width | bPOE | cdf | cdf_caption | cdf_image | cdf_image_alt | cdf2 | cf | cf2 | char | char2 | entropy | entropy2 | ES | fisher | fisher2 | intro | JSDdiv | KLDiv | kurtosis | kurtosis2 | mad | mad2 | aad | aad2 | mean | mean2 | median | median2 | mgf | mgf2 | mode | mode2 | moments | moments2 | name | notation | parameters | parameters2 | pdf | pdf_caption | pdf_image | pdf_image_alt | pdf2 | pgf | pgf2 | quantile | skewness | skewness2 | support | support2 | type | variance | variance2 }} e^{-\frac{(x - \mu)^2}{2\sigma^2}}</math>

 | cdf        = <math>\Phi\left(\frac{x-\mu}{\sigma}\right) = \frac{1}{2}\left[1 + \operatorname{erf}\left( \frac{x-\mu}{\sigma\sqrt{2}}\right)\right]</math>
 | quantile   = <math>\mu+\sigma\sqrt{2} \operatorname{erf}^{-1}(2p-1)</math>
 | mean       = <math>\mu</math>
 | median     = <math>\mu</math>
 | mode       = <math>\mu</math>
 | variance   = <math>\sigma^2</math>
 | mad        = <math>\sigma\sqrt{2}\,\operatorname{erf}^{-1}(1/2)</math>
 | aad        = <math display="inline">\sigma\sqrt{2/\pi}</math>
 | skewness   = <math>0</math>
 | kurtosis   = <math>0</math> 
 | entropy    = <math display="inline">\tfrac{1}{2} \log(2\pi e \sigma^2)</math>
 | mgf        = <math>\exp(\mu t + \sigma^2 t^2 / 2)</math>
 | char       = <math>\exp(i \mu t - \sigma^2 t^2 / 2)</math>
 | fisher     = <math>\mathcal{I}(\mu,\sigma) =\begin {pmatrix} 1/\sigma^2 & 0 \\ 0 & 2/\sigma^2\end{pmatrix}</math>

<math>\mathcal{I}(\mu,\sigma^2) =\begin {pmatrix} 1/\sigma^2 & 0 \\ 0 & 1/(2\sigma^4)\end{pmatrix}</math>

 | KLDiv      = <math>{ 1 \over 2 } \left\{ \left( \frac{\sigma_0}{\sigma_1} \right)^2 + \frac{(\mu_1 - \mu_0)^2}{\sigma_1^2} - 1 + \ln {\sigma_1^2 \over \sigma_0^2} \right\}</math>
 | ES         = <math>\mu + \sigma \frac{\frac{1}{\sqrt{2\pi}} e^{\frac{-\left(q_p\left(\frac{X-\mu}{\sigma}\right)\right)^2}{2}}}{1-p}</math><ref name="Norton-2019">Template:Cite journal</ref>

}} Template:Probability fundamentals

In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is<ref name="The Joy of Finite Mathematics">Template:Cite book</ref><ref name="Mathematics for Physical Science and Engineering">Template:Cite book</ref><ref>Template:Harvtxt and Template:Harvtxt give this definition with slightly different notation.</ref> <math display=block> f(x) = \frac{1}{\sqrt{2\pi\sigma^2} } e^{-\frac{(x-\mu)^2}{2\sigma^2}}\,. </math>

The parameter Template:Tmath is the mean or expectation of the distribution (and also its median and mode), while the parameter <math display=inline>\sigma^2</math> is the variance. The standard deviation of the distribution is Template:Tmath (sigma). A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.

Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.<ref>Normal Distribution, Gale Encyclopedia of Psychology</ref><ref>Template:Harvtxt</ref> Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes, such as measurement errors, often have distributions that are nearly normal.<ref>Lyon, A. (2014). Why are Normal Distributions Normal?, The British Journal for the Philosophy of Science.</ref>

Moreover, Gaussian distributions have some unique properties that are valuable in analytic studies. For instance, any linear combination of a fixed collection of independent normal deviates is a normal deviate. Many results and methods, such as propagation of uncertainty and least squares<ref>Template:Cite book</ref> parameter fitting, can be derived analytically in explicit form when the relevant variables are normally distributed.

A normal distribution is sometimes informally called a bell curve.<ref name="www.mathsisfun.com">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> However, many other distributions are bell-shaped (such as the Cauchy, Student's t, and logistic distributions). (For other names, see Naming.)

The univariate probability distribution is generalized for vectors in the multivariate normal distribution and for matrices in the matrix normal distribution.

DefinitionsEdit

Standard normal distributionEdit

The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution. This is a special case when <math display=inline>\mu=0</math> and <math display=inline>\sigma^2 =1</math>, and it is described by this probability density function (or density):<ref>Template:Harvtxt explicitly defines the standard normal distribution. In contrast, Template:Harvtxt explicitly defines the standard normal curve (p. 33) and introduces the term standard normal distribution (p. 69).</ref> <math display="block">\varphi(z) = \frac{e^{-z^2/2}}{\sqrt{2\pi}}\,.</math> The variable Template:Tmath has a mean of 0 and a variance and standard deviation of 1. The density <math display=inline>\varphi(z)</math> has its peak <math display="inline">\frac{1}{\sqrt{2\pi}}</math> at <math display=inline>z=0</math> and inflection points at <math display=inline>z=+1</math> and Template:Tmath.

Although the density above is most commonly known as the standard normal, a few authors have used that term to describe other versions of the normal distribution. Carl Friedrich Gauss, for example, once defined the standard normal as <math display=block>\varphi(z) = \frac{e^{-z^2}}{\sqrt\pi},</math> which has a variance of Template:Tmath, and Stephen Stigler<ref>Template:Harvtxt</ref> once defined the standard normal as <math display=block>\varphi(z) = e^{-\pi z^2},</math> which has a simple functional form and a variance of <math display=inline>\sigma^2 = \frac {1}{2\pi}.</math>

General normal distributionEdit

Every normal distribution is a version of the standard normal distribution, whose domain has been stretched by a factor Template:Tmath (the standard deviation) and then translated by Template:Tmath (the mean value):

<math display=block> f(x \mid \mu, \sigma^2) =\frac 1 \sigma \varphi\left(\frac{x-\mu} \sigma \right)\,. </math>

The probability density must be scaled by <math display=inline>1/\sigma</math> so that the integral is still 1.

If Template:Tmath is a standard normal deviate, then <math display=inline>X=\sigma Z + \mu</math> will have a normal distribution with expected value Template:Tmath and standard deviation Template:Tmath. This is equivalent to saying that the standard normal distribution Template:Tmath can be scaled/stretched by a factor of Template:Tmath and shifted by Template:Tmath to yield a different normal distribution, called Template:Tmath. Conversely, if Template:Tmath is a normal deviate with parameters Template:Tmath and <math display=inline>\sigma^2</math>, then this Template:Tmath distribution can be re-scaled and shifted via the formula <math display=inline>Z=(X-\mu)/\sigma</math> to convert it to the standard normal distribution. This variate is also called the standardized form of Template:Tmath.

NotationEdit

The probability density of the standard Gaussian distribution (standard normal distribution, with zero mean and unit variance) is often denoted with the Greek letter Template:Tmath (phi).<ref>Template:Harvtxt</ref> The alternative form of the Greek letter phi, Template:Tmath, is also used quite often.

The normal distribution is often referred to as <math display=inline>N(\mu,\sigma^2)</math> or Template:Tmath.<ref>Template:Harvtxt</ref> Thus when a random variable Template:Tmath is normally distributed with mean Template:Tmath and standard deviation Template:Tmath, one may write

<math display=block>X \sim \mathcal{N}(\mu,\sigma^2).</math>

Alternative parameterizationsEdit

Some authors advocate using the precision Template:Tmath as the parameter defining the width of the distribution, instead of the standard deviation Template:Tmath or the variance Template:Tmath. The precision is normally defined as the reciprocal of the variance, Template:Tmath.<ref>Template:Harvtxt</ref> The formula for the distribution then becomes

This choice is claimed to have advantages in numerical computations when Template:Tmath is very close to zero, and simplifies formulas in some contexts, such as in the Bayesian inference of variables with multivariate normal distribution.

Alternatively, the reciprocal of the standard deviation <math display=inline>\tau'=1/\sigma</math> might be defined as the precision, in which case the expression of the normal distribution becomes

According to Stigler, this formulation is advantageous because of a much simpler and easier-to-remember formula, and simple approximate formulas for the quantiles of the distribution.

Normal distributions form an exponential family with natural parameters <math display=inline>\textstyle\theta_1=\frac{\mu}{\sigma^2}</math> and <math display=inline>\textstyle\theta_2=\frac{-1}{2\sigma^2}</math>, and natural statistics x and x². The dual expectation parameters for normal distribution are Template:Math and Template:Math.

Cumulative distribution functionEdit

The cumulative distribution function (CDF) of the standard normal distribution, usually denoted with the capital Greek letter Template:Tmath, is the integral

<math display=block>\Phi(x) = \frac 1 {\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2/2} \, dt\,.</math>

Error functionEdit

The related error function <math display=inline>\operatorname{erf}(x)</math> gives the probability of a random variable, with normal distribution of mean 0 and variance 1/2 falling in the range Template:Tmath. That is:

<math display=block>\operatorname{erf}(x) = \frac 1 {\sqrt\pi} \int_{-x}^x e^{-t^2} \, dt = \frac 2 {\sqrt\pi} \int_0^x e^{-t^2} \, dt\,.</math>

These integrals cannot be expressed in terms of elementary functions, and are often said to be special functions. However, many numerical approximations are known; see below for more.

The two functions are closely related, namely

<math display=block>\Phi(x) = \frac{1}{2} \left[1 + \operatorname{erf}\left( \frac x {\sqrt 2} \right) \right]\,.</math>

For a generic normal distribution with density Template:Tmath, mean Template:Tmath and variance <math display=inline>\sigma^2</math>, the cumulative distribution function is

<math display="block"> F(x) = \Phi{\left(\frac{x-\mu} \sigma \right)} = \frac{1}{2} \left[1 + \operatorname{erf}\left(\frac{x-\mu}{\sigma \sqrt 2 }\right)\right]\,. </math>

The complement of the standard normal cumulative distribution function, <math display=inline>Q(x) = 1 - \Phi(x)</math>, is often called the Q-function, especially in engineering texts.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> It gives the probability that the value of a standard normal random variable Template:Tmath will exceed Template:Tmath: Template:Tmath. Other definitions of the Template:Tmath-function, all of which are simple transformations of Template:Tmath, are also used occasionally.<ref>{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:NormalDistributionFunction%7CNormalDistributionFunction.html}} |title = Normal Distribution Function |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</ref>

The graph of the standard normal cumulative distribution function Template:Tmath has 2-fold rotational symmetry around the point (0,1/2); that is, Template:Tmath. Its antiderivative (indefinite integral) can be expressed as follows: <math display=block>\int \Phi(x)\, dx = x\Phi(x) + \varphi(x) + C.</math>

The cumulative distribution function of the standard normal distribution can be expanded by integration by parts into a series:

<math display=block>\Phi(x)=\frac{1}{2} + \frac{1}{\sqrt{2\pi}}\cdot e^{-x^2/2} \left[x + \frac{x^3}{3} + \frac{x^5}{3\cdot 5} + \cdots + \frac{x^{2n+1}}{(2n+1)!!} + \cdots\right]\,.</math>

where <math display=inline>!!</math> denotes the double factorial.

An asymptotic expansion of the cumulative distribution function for large x can also be derived using integration by parts. For more, see Template:Slink.<ref>Template:AS ref</ref>

A quick approximation to the standard normal distribution's cumulative distribution function can be found by using a Taylor series approximation:

<math display=block>\Phi(x) \approx \frac{1}{2}+\frac{1}{\sqrt{2\pi}} \sum_{k=0}^n \frac{(-1)^k x^{(2k+1)}}{2^k k! (2k+1)}\,.</math>

Recursive computation with Taylor series expansionEdit

The recursive nature of the <math display=inline>e^{ax^2}</math>family of derivatives may be used to easily construct a rapidly converging Taylor series expansion using recursive entries about any point of known value of the distribution,<math display=inline>\Phi(x_0)</math>:

<math display=block>\Phi(x) = \sum_{n=0}^\infty \frac{\Phi^{(n)}(x_0)}{n!}(x-x_0)^n\,,</math>

where:

<math display=block>\begin{align} \Phi^{(0)}(x_0) &= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x_0}e^{-t^2/2}\,dt \\ \Phi^{(1)}(x_0) &= \frac{1}{\sqrt{2\pi}}e^{-x_0^2/2} \\ \Phi^{(n)}(x_0) &= -\left(x_0\Phi^{(n-1)}(x_0)+(n-2)\Phi^{(n-2)}(x_0)\right), & n \geq 2\,. \end{align}</math>

Using the Taylor series and Newton's method for the inverse functionEdit

An application for the above Taylor series expansion is to use Newton's method to reverse the computation. That is, if we have a value for the cumulative distribution function, <math display=inline>\Phi(x)</math>, but do not know the x needed to obtain the <math display=inline>\Phi(x)</math>, we can use Newton's method to find x, and use the Taylor series expansion above to minimize the number of computations. Newton's method is ideal to solve this problem because the first derivative of <math display=inline>\Phi(x)</math>, which is an integral of the normal standard distribution, is the normal standard distribution, and is readily available to use in the Newton's method solution.

To solve, select a known approximate solution, <math display=inline>x_0</math>, to the desired Template:Tmath. <math display=inline>x_0</math> may be a value from a distribution table, or an intelligent estimate followed by a computation of <math display=inline>\Phi(x_0)</math> using any desired means to compute. Use this value of <math display=inline>x_0</math> and the Taylor series expansion above to minimize computations.

Repeat the following process until the difference between the computed <math display=inline>\Phi(x_{n})</math> and the desired Template:Tmath, which we will call <math display=inline>\Phi(\text{desired})</math>, is below a chosen acceptably small error, such as 10⁻⁵, 10⁻¹⁵, etc.:

<math display=block>x_{n+1} = x_n - \frac{\Phi(x_n,x_0,\Phi(x_0))-\Phi(\text{desired})}{\Phi'(x_n)}\,,</math>

where

<math display=inline>\Phi(x,x_0,\Phi(x_0))</math> is the <math display=inline>\Phi(x)</math> from a Taylor series solution using <math display=inline>x_0</math> and <math display=inline>\Phi(x_0)</math>

When the repeated computations converge to an error below the chosen acceptably small value, x will be the value needed to obtain a <math display=inline>\Phi(x)</math> of the desired value, Template:Tmath.

Standard deviation and coverageEdit

Template:Further

File:Standard deviation diagram.svg

For the normal distribution, the values less than one standard deviation from the mean account for 68.27% of the set; while two standard deviations from the mean account for 95.45%; and three standard deviations account for 99.73%.

About 68% of values drawn from a normal distribution are within one standard deviation σ from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations.<ref name="www.mathsisfun.com" /> This fact is known as the 68–95–99.7 (empirical) rule, or the 3-sigma rule.

More precisely, the probability that a normal deviate lies in the range between <math display=inline>\mu-n\sigma</math> and <math display=inline>\mu+n\sigma</math> is given by <math display=block> F(\mu+n\sigma) - F(\mu-n\sigma) = \Phi(n)-\Phi(-n) = \operatorname{erf} \left(\frac{n}{\sqrt{2}}\right). </math> To 12 significant digits, the values for <math display=inline>n=1,2,\ldots , 6</math> are:

Template:Tmath

<math display=inline>p= F(\mu+n\sigma) - F(\mu-n\sigma)</math>

1

2

3

4

5

6

For large Template:Tmath, one can use the approximation <math display=inline>1 - p \approx \frac{e^{-n^2/2}}{n\sqrt{\pi/2}}</math>.

Quantile functionEdit

Template:Further

The quantile function of a distribution is the inverse of the cumulative distribution function. The quantile function of the standard normal distribution is called the probit function, and can be expressed in terms of the inverse error function: <math display=block> \Phi^{-1}(p) = \sqrt2\operatorname{erf}^{-1}(2p - 1), \quad p\in(0,1). </math> For a normal random variable with mean Template:Tmath and variance <math display=inline>\sigma^2</math>, the quantile function is <math display=block> F^{-1}(p) = \mu + \sigma\Phi^{-1}(p) = \mu + \sigma\sqrt 2 \operatorname{erf}^{-1}(2p - 1), \quad p\in(0,1). </math> The quantile <math display=inline>\Phi^{-1}(p)</math> of the standard normal distribution is commonly denoted as Template:Tmath. These values are used in hypothesis testing, construction of confidence intervals and Q–Q plots. A normal random variable Template:Tmath will exceed <math display=inline>\mu + z_p\sigma</math> with probability <math display=inline>1-p</math>, and will lie outside the interval <math display=inline>\mu \pm z_p\sigma</math> with probability Template:Tmath. In particular, the quantile <math display=inline>z_{0.975}</math> is 1.96; therefore a normal random variable will lie outside the interval <math display=inline>\mu \pm 1.96\sigma</math> in only 5% of cases.

The following table gives the quantile <math display=inline>z_p</math> such that Template:Tmath will lie in the range <math display=inline>\mu \pm z_p\sigma</math> with a specified probability Template:Tmath. These values are useful to determine tolerance interval for sample averages and other statistical estimators with normal (or asymptotically normal) distributions.<ref>Template:Cite book</ref> The following table shows <math display=inline>\sqrt 2 \operatorname{erf}^{-1}(p)=\Phi^{-1}\left(\frac{p+1}{2}\right)</math>, not <math display=inline>\Phi^{-1}(p)</math> as defined above.

Template:Tmath	<math display=inline>z_p</math>	Template:Tmath	<math display=inline>z_p</math>
0.80	Template:Val	0.999	Template:Val
0.90	Template:Val	0.9999	Template:Val
0.95	Template:Val	0.99999	Template:Val
0.98	Template:Val	0.999999	Template:Val
0.99	Template:Val	0.9999999	Template:Val
0.995	Template:Val	0.99999999	Template:Val
0.998	Template:Val	0.999999999	Template:Val

For small Template:Tmath, the quantile function has the useful asymptotic expansion <math display=inline>\Phi^{-1}(p)=-\sqrt{\ln\frac{1}{p^2}-\ln\ln\frac{1}{p^2}-\ln(2\pi)}+\mathcal{o}(1).</math>Template:Citation needed

PropertiesEdit

The normal distribution is the only distribution whose cumulants beyond the first two (i.e., other than the mean and variance) are zero. It is also the continuous distribution with the maximum entropy for a specified mean and variance.Template:Sfnp<ref>Template:Cite journal</ref> Geary has shown, assuming that the mean and variance are finite, that the normal distribution is the only distribution where the mean and variance calculated from a set of independent draws are independent of each other.<ref name="Geary RC">Geary RC(1936) The distribution of the "Student's ratio for the non-normal samples". Supplement to the Journal of the Royal Statistical Society 3 (2): 178–184</ref><ref>Template:Cite Q</ref>

The normal distribution is a subclass of the elliptical distributions. The normal distribution is symmetric about its mean, and is non-zero over the entire real line. As such it may not be a suitable model for variables that are inherently positive or strongly skewed, such as the weight of a person or the price of a share. Such variables may be better described by other distributions, such as the log-normal distribution or the Pareto distribution.

The value of the normal density is practically zero when the value Template:Tmath lies more than a few standard deviations away from the mean (e.g., a spread of three standard deviations covers all but 0.27% of the total distribution). Therefore, it may not be an appropriate model when one expects a significant fraction of outliers—values that lie many standard deviations away from the mean—and least squares and other statistical inference methods that are optimal for normally distributed variables often become highly unreliable when applied to such data. In those cases, a more heavy-tailed distribution should be assumed and the appropriate robust statistical inference methods applied.

The Gaussian distribution belongs to the family of stable distributions which are the attractors of sums of independent, identically distributed distributions whether or not the mean or variance is finite. Except for the Gaussian which is a limiting case, all stable distributions have heavy tails and infinite variance. It is one of the few distributions that are stable and that have probability density functions that can be expressed analytically, the others being the Cauchy distribution and the Lévy distribution.

Symmetries and derivativesEdit

The normal distribution with density <math display=inline>f(x)</math> (mean Template:Tmath and variance <math display=inline>\sigma^2 > 0</math>) has the following properties:

It is symmetric around the point <math display=inline>x=\mu,</math> which is at the same time the mode, the median and the mean of the distribution.<ref name="Patel">Template:Harvtxt</ref>
It is unimodal: its first derivative is positive for <math display=inline>x<\mu,</math> negative for <math display=inline>x>\mu,</math> and zero only at <math display=inline>x=\mu.</math>
The area bounded by the curve and the Template:Tmath-axis is unity (i.e. equal to one).
Its first derivative is <math display=inline>f'(x)=-\frac{x-\mu}{\sigma^2} f(x).</math>
Its second derivative is <math display=inline>f(x) = \frac{(x-\mu)^2 - \sigma^2}{\sigma^4} f(x).</math>
Its density has two inflection points (where the second derivative of Template:Tmath is zero and changes sign), located one standard deviation away from the mean, namely at <math display=inline>x=\mu-\sigma</math> and <math display=inline>x=\mu+\sigma.</math><ref name="Patel" />
Its density is log-concave.<ref name="Patel" />
Its density is infinitely differentiable, indeed supersmooth of order 2.<ref>Template:Harvtxt</ref>

Furthermore, the density Template:Tmath of the standard normal distribution (i.e. <math display=inline>\mu=0</math> and <math display=inline>\sigma=1</math>) also has the following properties:

Its first derivative is <math display=inline>\varphi'(x)=-x\varphi(x).</math>
Its second derivative is <math display=inline>\varphi(x)=(x^2-1)\varphi(x)</math>
More generally, its Template:Mvarth derivative is <math display=inline>\varphi^{(n)}(x) = (-1)^n\operatorname{He}_n(x)\varphi(x),</math> where <math display=inline>\operatorname{He}_n(x)</math> is the Template:Mvarth (probabilist) Hermite polynomial.<ref>Template:Harvtxt</ref>
The probability that a normally distributed variable Template:Tmath with known Template:Tmath and <math display=inline>\sigma^2</math> is in a particular set, can be calculated by using the fact that the fraction <math display=inline>Z = (X-\mu)/\sigma</math> has a standard normal distribution.

MomentsEdit

Template:See also The plain and absolute moments of a variable Template:Tmath are the expected values of <math display=inline>X^p</math> and <math display=inline>|X|^p</math>, respectively. If the expected value Template:Tmath of Template:Tmath is zero, these parameters are called central moments; otherwise, these parameters are called non-central moments. Usually we are interested only in moments with integer order Template:Tmath.

If Template:Tmath has a normal distribution, the non-central moments exist and are finite for any Template:Tmath whose real part is greater than −1. For any non-negative integer Template:Tmath, the plain central moments are:<ref>Template:Cite book</ref> <math display=block>

   \operatorname{E}\left[(X-\mu)^p\right] =
     \begin{cases}
       0 & \text{if }p\text{ is odd,} \\
       \sigma^p (p-1)!! & \text{if }p\text{ is even.}
     \end{cases}

</math> Here <math display=inline>n!!</math> denotes the double factorial, that is, the product of all numbers from Template:Tmath to 1 that have the same parity as <math display=inline>n.</math>

The central absolute moments coincide with plain moments for all even orders, but are nonzero for odd orders. For any non-negative integer <math display=inline>p,</math>

<math display=block>\begin{align}

  \operatorname{E}\left[|X - \mu|^p\right] &= \sigma^p (p-1)!! \cdot \begin{cases}
    \sqrt{\frac{2}{\pi}} & \text{if }p\text{ is odd} \\
    1 & \text{if }p\text{ is even}
  \end{cases} \\
  &= \sigma^p \cdot \frac{2^{p/2}\Gamma\left(\frac{p+1} 2 \right)}{\sqrt\pi}.
\end{align}</math>

The last formula is valid also for any non-integer <math display=inline>p>-1.</math> When the mean <math display=inline>\mu \ne 0,</math> the plain and absolute moments can be expressed in terms of confluent hypergeometric functions <math display=inline>{}_1F_1</math> and <math display=inline>U.</math><ref>Template:Cite arXiv</ref>

<math display="block">\begin{align}

  \operatorname{E}\left[X^p\right] &= \sigma^p\cdot {\left(-i\sqrt 2\right)}^p \, U{\left(-\frac{p}{2}, \frac{1}{2}, -\frac{\mu^2}{2\sigma^2}\right)}, \\
  \operatorname{E}\left[|X|^p \right] &= \sigma^p \cdot 2^{p/2} \frac {\Gamma{\left(\frac{1+p} 2\right)}}{\sqrt\pi} \, {}_1F_1{\left( -\frac{p}{2}, \frac{1}{2}, -\frac{\mu^2}{2\sigma^2} \right)}.
\end{align}</math>

These expressions remain valid even if Template:Tmath is not an integer. See also generalized Hermite polynomials.

Order	Non-central moment, <math>\operatorname{E}\left[X^p\right]</math>	Central moment, <math>\operatorname{E}\left[(X-\mu)^p\right]</math>
1	Template:Tmath	Template:Tmath
2	<math display=inline>\mu^2+\sigma^2</math>	<math display=inline>\sigma^2</math>
3	<math display=inline>\mu^3+3\mu\sigma^2</math>	Template:Tmath
4	<math display=inline>\mu^4+6\mu^2\sigma^2+3\sigma^4</math>	<math display=inline>3\sigma^4</math>
5	<math display=inline>\mu^5+10\mu^3\sigma^2+15\mu\sigma^4</math>	Template:Tmath
6	<math display=inline>\mu^6+15\mu^4\sigma^2+45\mu^2\sigma^4+15\sigma^6</math>	<math display=inline>15\sigma^6</math>
7	<math display=inline>\mu^7+21\mu^5\sigma^2+105\mu^3\sigma^4+105\mu\sigma^6</math>	Template:Tmath
8	<math display=inline>\mu^8+28\mu^6\sigma^2+210\mu^4\sigma^4+420\mu^2\sigma^6+105\sigma^8</math>	<math display=inline>105\sigma^8</math>

The expectation of Template:Tmath conditioned on the event that Template:Tmath lies in an interval <math display=inline>[a,b]</math> is given by <math display=block>\operatorname{E}\left[X \mid a<X<b \right] = \mu - \sigma^2\frac{f(b)-f(a)}{F(b)-F(a)}\,,</math> where Template:Tmath and Template:Tmath respectively are the density and the cumulative distribution function of Template:Tmath. For <math display=inline>b=\infty</math> this is known as the inverse Mills ratio. Note that above, density Template:Tmath of Template:Tmath is used instead of standard normal density as in inverse Mills ratio, so here we have <math display=inline>\sigma^2</math> instead of Template:Tmath.

Fourier transform and characteristic functionEdit

The Fourier transform of a normal density Template:Tmath with mean Template:Tmath and variance <math display=inline>\sigma^2</math> is<ref>Template:Harvtxt</ref>

<math display=block> \hat f(t) = \int_{-\infty}^\infty f(x)e^{-itx} \, dx = e^{-i\mu t} e^{- \frac12 (\sigma t)^2}\,, </math>

where Template:Tmath is the imaginary unit. If the mean <math display=inline>\mu=0</math>, the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the frequency domain, with mean 0 and variance Template:Tmath. In particular, the standard normal distribution Template:Tmath is an eigenfunction of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable Template:Tmath is closely connected to the characteristic function <math display=inline>\varphi_X(t)</math> of that variable, which is defined as the expected value of <math display=inline>e^{itX}</math>, as a function of the real variable Template:Tmath (the frequency parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable Template:Tmath.<ref>Template:Harvtxt</ref> The relation between both is: <math display=block>\varphi_X(t) = \hat f(-t)\,.</math>

Moment- and cumulant-generating functionsEdit

The moment generating function of a real random variable Template:Tmath is the expected value of <math display=inline>e^{tX}</math>, as a function of the real parameter Template:Tmath. For a normal distribution with density Template:Tmath, mean Template:Tmath and variance <math display=inline>\sigma^2</math>, the moment generating function exists and is equal to

<math display=block>M(t) = \operatorname{E}\left[e^{tX}\right] = \hat f(it) = e^{\mu t} e^{\sigma^2 t^2/2}\,.</math> For any Template:Tmath, the coefficient of Template:Tmath in the moment generating function (expressed as an exponential power series in Template:Tmath) is the normal distribution's expected value Template:Tmath.

The cumulant generating function is the logarithm of the moment generating function, namely

<math display=block>g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.</math>

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in Template:Tmath, only the first two cumulants are nonzero, namely the mean Template:Tmath and the variance Template:Tmath.

Some authors prefer to instead work with the characteristic function Template:Math and Template:Math.

Stein operator and classEdit

Within Stein's method the Stein operator and class of a random variable <math display=inline>X \sim \mathcal{N}(\mu, \sigma^2)</math> are <math display=inline>\mathcal{A}f(x) = \sigma^2 f'(x) - (x-\mu)f(x)</math> and <math display=inline>\mathcal{F}</math> the class of all absolutely continuous functions Template:Tmath such that Template:Tmath.

Zero-variance limitEdit

In the limit when <math display=inline>\sigma^2</math> tends to zero, the probability density <math display=inline>f(x)</math> eventually tends to zero at any <math display=inline>x\ne \mu</math>, but grows without limit if <math display=inline>x = \mu</math>, while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function when Template:Tmath.

However, one can define the normal distribution with zero variance as a generalized function; specifically, as a Dirac delta function Template:Tmath translated by the mean Template:Tmath, that is <math display=inline>f(x)=\delta(x-\mu).</math> Its cumulative distribution function is then the Heaviside step function translated by the mean Template:Tmath, namely <math display=block>F(x) = \begin{cases}

 0 & \text{if }x < \mu \\
 1 & \text{if }x \geq \mu\,.

\end{cases} </math>

Maximum entropyEdit

Of all probability distributions over the reals with a specified finite mean Template:Tmath and finite variance Template:Tmath, the normal distribution <math display=inline>N(\mu,\sigma^2)</math> is the one with maximum entropy.Template:Sfnp To see this, let Template:Tmath be a continuous random variable with probability density Template:Tmath. The entropy of Template:Tmath is defined as<ref>Template:Cite book</ref><ref>Template:Cite book</ref><ref>O'Hagan, A. (1994) Kendall's Advanced Theory of statistics, Vol 2B, Bayesian Inference, Edward Arnold. Template:Isbn (Section 5.40)</ref> <math display=block> H(X) = - \int_{-\infty}^\infty f(x)\ln f(x)\, dx\,, </math>

where <math display=inline>f(x)\log f(x)</math> is understood to be zero whenever Template:Tmath. This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using variational calculus. A function with three Lagrange multipliers is defined:

<math display=block> L=-\int_{-\infty}^\infty f(x)\ln f(x)\,dx-\lambda_0\left(1-\int_{-\infty}^\infty f(x)\,dx\right)-\lambda_1\left(\mu-\int_{-\infty}^\infty f(x)x\,dx\right)-\lambda_2\left(\sigma^2-\int_{-\infty}^\infty f(x)(x-\mu)^2\,dx\right)\,. </math>

At maximum entropy, a small variation <math display=inline>\delta f(x)</math> about <math display=inline>f(x)</math> will produce a variation <math display=inline>\delta L</math> about Template:Tmath which is equal to 0:

<math display=block> 0=\delta L=\int_{-\infty}^\infty \delta f(x)\left(-\ln f(x) -1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,dx\,. </math>

Since this must hold for any small Template:Tmath, the factor multiplying Template:Tmath must be zero, and solving for Template:Tmath yields:

<math display=block>f(x)=\exp\left(-1+\lambda_0+\lambda_1 x+\lambda_2(x-\mu)^2\right)\,.</math>

The Lagrange constraints that Template:Tmath is properly normalized and has the specified mean and variance are satisfied if and only if Template:Tmath, Template:Tmath, and Template:Tmath are chosen so that <math display=block> f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\,. </math> The entropy of a normal distribution <math display=inline>X \sim N(\mu,\sigma^2)</math> is equal to <math display=block> H(X)=\tfrac{1}{2}(1+\ln 2\sigma^2\pi)\,, </math> which is independent of the mean Template:Tmath.

Other propertiesEdit

Template:Ordered list \exp\left(-\frac{1}{4}\frac{(\mu_1-\mu_2)^2}{\sigma_1^2+\sigma_2^2}\right) </math> | 4 = The Fisher information matrix for a normal distribution w.r.t. Template:Tmath and <math display=inline>\sigma^2</math> is diagonal and takes the form <math display=block>

   \mathcal I (\mu, \sigma^2) = \begin{pmatrix} \frac{1}{\sigma^2} & 0 \\ 0 & \frac{1}{2\sigma^4} \end{pmatrix}

</math> | 5 = The conjugate prior of the mean of a normal distribution is another normal distribution.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Specifically, if <math display=inline>x_1, \ldots, x_n</math> are iid <math display=inline>\sim N(\mu, \sigma^2)</math> and the prior is <math display=inline>\mu \sim N(\mu_0 , \sigma^2_0)</math>, then the posterior distribution for the estimator of Template:Tmath will be <math display=block>

   \mu \mid x_1,\ldots,x_n \sim \mathcal{N}\left( \frac{\frac{\sigma^2}{n}\mu_0 + \sigma_0^2\bar{x}}{\frac{\sigma^2}{n}+\sigma_0^2},\left( \frac{n}{\sigma^2} + \frac{1}{\sigma_0^2} \right)^{-1} \right)

</math> | 6 = The family of normal distributions not only forms an exponential family (EF), but in fact forms a natural exponential family (NEF) with quadratic variance function (NEF-QVF). Many properties of normal distributions generalize to properties of NEF-QVF distributions, NEF distributions, or EF distributions generally. NEF-QVF distributions comprises 6 families, including Poisson, Gamma, binomial, and negative binomial distributions, while many of the common families studied in probability and statistics are NEF or EF. | 7 = In information geometry, the family of normal distributions forms a statistical manifold with constant curvature Template:Tmath. The same family is flat with respect to the (±1)-connections <math display=inline>\nabla^{(e)}</math> and <math display=inline>\nabla^{(m)}</math>.<ref>Template:Harvtxt</ref> | 8 = If <math display=inline>X_1, \dots, X_n</math> are distributed according to <math display=inline>N(0, \sigma^2)</math>, then <math display=inline>E[\max_i X_i ] \leq \sigma\sqrt{2\ln n}</math>. Note that there is no assumption of independence.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> }}

Related distributionsEdit

Central limit theoremEdit

File:De moivre-laplace.gif

As the number of discrete events increases, the function begins to resemble a normal distribution.

File:Dice sum central limit theorem.svg

Comparison of probability density functions, Template:Math for the sum of Template:Mvar fair 6-sided dice to show their convergence to a normal distribution with increasing Template:Mvar, in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where <math display=inline>X_1,\ldots ,X_n</math> are independent and identically distributed random variables with the same arbitrary distribution, zero mean, and variance <math display=inline>\sigma^2</math> and Template:Tmath is their mean scaled by <math display=inline>\sqrt{n}</math> <math display=block>Z = \sqrt{n}\left(\frac{1}{n}\sum_{i=1}^n X_i\right)</math> Then, as Template:Tmath increases, the probability distribution of Template:Tmath will tend to the normal distribution with zero mean and variance Template:Tmath.

The theorem can be extended to variables <math display=inline>(X_i)</math> that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.

Many test statistics, scores, and estimators encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.

The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:

The binomial distribution <math display=inline>B(n,p)</math> is approximately normal with mean <math display=inline>np</math> and variance <math display=inline>np(1-p)</math> for large Template:Tmath and for Template:Tmath not too close to 0 or 1.
The Poisson distribution with parameter Template:Tmath is approximately normal with mean Template:Tmath and variance Template:Tmath, for large values of Template:Tmath.<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

The chi-squared distribution <math display=inline>\chi^2(k)</math> is approximately normal with mean Template:Tmath and variance <math display=inline>2k</math>, for large Template:Tmath.
The Student's t-distribution <math display=inline>t(\nu)</math> is approximately normal with mean 0 and variance 1 when Template:Tmath is large.

Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.

A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.

This theorem can also be used to justify modeling the sum of many uniform noise sources as Gaussian noise. See AWGN.

Operations and functions of normal variablesEdit

File:Probabilities of functions of normal vectors.png

a: Probability density of a function Template:Math of a normal variable Template:Mvar with Template:Math and Template:Math. b: Probability density of a function Template:Mvar of two normal variables Template:Mvar and Template:Mvar, where Template:Math, Template:Math, Template:Math, Template:Math, and Template:Math. c: Heat map of the joint probability density of two functions of two correlated normal variables Template:Mvar and Template:Mvar, where Template:Math, Template:Math, Template:Math, Template:Math, and Template:Math. d: Probability density of a function Template:Math of four iid standard normal variables. These are computed by the numerical method of ray-tracing.<ref name="Das-2021" />

The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing<ref name="Das-2021">Template:Cite journal</ref> (Matlab code). In the following sections we look at some special cases.

Operations on a single normal variableEdit

If Template:Tmath is distributed normally with mean Template:Tmath and variance <math display=inline>\sigma^2</math>, then

<math display=inline>aX+b</math>, for any real numbers Template:Tmath and Template:Tmath, is also normally distributed, with mean <math display=inline>a\mu+b</math> and variance <math display=inline>a^2\sigma^2</math>. That is, the family of normal distributions is closed under linear transformations.
The exponential of Template:Tmath is distributed log-normally: <math display=inline>e^X \sim \ln(N(\mu, \sigma^2))</math>.
The standard sigmoid of Template:Tmath is logit-normally distributed: <math display=inline>\sigma(X) \sim P( \mathcal{N}(\mu,\,\sigma^2) )</math>.
The absolute value of Template:Tmath has folded normal distribution: <math display=inline>Template:\left</math>. If <math display=inline>\mu = 0</math> this is known as the half-normal distribution.
The absolute value of normalized residuals, <math display=inline>|X - \mu| / \sigma</math>, has chi distribution with one degree of freedom: <math display=inline>|X - \mu| / \sigma \sim \chi_1</math>.
The square of <math display=inline>X/\sigma</math> has the noncentral chi-squared distribution with one degree of freedom: <math display=inline>X^2 / \sigma^2 \sim \chi_1^2(\mu^2 / \sigma^2)</math>. If <math display=inline>\mu = 0</math>, the distribution is called simply chi-squared.
The log-likelihood of a normal variable Template:Tmath is simply the log of its probability density function: <math display=block>\ln p(x)= -\frac{1}{2} \left(\frac{x-\mu}{\sigma} \right)^2 -\ln \left(\sigma \sqrt{2\pi} \right).</math> Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.
The distribution of the variable Template:Tmath restricted to an interval <math display=inline>[a, b]</math> is called the truncated normal distribution.
<math display=inline>(X - \mu)^{-2}</math> has a Lévy distribution with location 0 and scale <math display=inline>\sigma^{-2}</math>.

Operations on two independent normal variablesEdit

If <math display=inline>X_1</math> and <math display=inline>X_2</math> are two independent normal random variables, with means <math display=inline>\mu_1</math>, <math display=inline>\mu_2</math> and variances <math display=inline>\sigma_1^2</math>, <math display=inline>\sigma_2^2</math>, then their sum <math display=inline>X_1 + X_2</math> will also be normally distributed,^[proof] with mean <math display=inline>\mu_1 + \mu_2</math> and variance <math display=inline>\sigma_1^2 + \sigma_2^2</math>.
In particular, if Template:Tmath and Template:Tmath are independent normal deviates with zero mean and variance <math display=inline>\sigma^2</math>, then <math display=inline>X + Y</math> and <math display=inline>X - Y</math> are also independent and normally distributed, with zero mean and variance <math display=inline>2\sigma^2</math>. This is a special case of the polarization identity.<ref>Template:Harvtxt</ref>
If <math display=inline>X_1</math>, <math display=inline>X_2</math> are two independent normal deviates with mean Template:Tmath and variance <math display=inline>\sigma^2</math>, and Template:Tmath, Template:Tmath are arbitrary real numbers, then the variable <math display=block>

   X_3 = \frac{aX_1 + bX_2 - (a+b)\mu}{\sqrt{a^2+b^2}} + \mu

</math> is also normally distributed with mean Template:Tmath and variance <math display=inline>\sigma^2</math>. It follows that the normal distribution is stable (with exponent <math display=inline>\alpha=2</math>).

If <math display=inline>X_k \sim \mathcal N(m_k, \sigma_k^2)</math>, <math display=inline>k \in \{ 0, 1 \}</math> are normal distributions, then their normalized geometric mean <math display=inline>\frac{1}{\int_{\R^n} X_0^{\alpha}(x) X_1^{1 - \alpha}(x) \, \text{d}x} X_0^{\alpha} X_1^{1 - \alpha}</math> is a normal distribution <math display=inline>\mathcal N(m_{\alpha}, \sigma_{\alpha}^2)</math> with <math display=inline>m_{\alpha} = \frac{\alpha m_0 \sigma_1^2 + (1 - \alpha) m_1 \sigma_0^2}{\alpha \sigma_1^2 + (1 - \alpha) \sigma_0^2}</math> and <math display=inline>\sigma_{\alpha}^2 = \frac{\sigma_0^2 \sigma_1^2}{\alpha \sigma_1^2 + (1 - \alpha) \sigma_0^2}</math>.

Operations on two independent standard normal variablesEdit

If <math display=inline>X_1</math> and <math display=inline>X_2</math> are two independent standard normal random variables with mean 0 and variance 1, then

Their sum and difference is distributed normally with mean zero and variance two: <math display=inline>X_1 \pm X_2 \sim \mathcal{N}(0, 2)</math>.
Their product <math display=inline>Z = X_1 X_2</math> follows the product distribution<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref> with density function <math display=inline>f_Z(z) = \pi^{-1} K_0(|z|)</math> where <math display=inline>K_0</math> is the modified Bessel function of the second kind. This distribution is symmetric around zero, unbounded at <math display=inline>z = 0</math>, and has the characteristic function <math display=inline>\phi_Z(t) = (1 + t^2)^{-1/2}</math>.

Their ratio follows the standard Cauchy distribution: <math display=inline>X_1/ X_2 \sim \operatorname{Cauchy}(0, 1)</math>.
Their Euclidean norm <math display=inline>\sqrt{X_1^2 + X_2^2}</math> has the Rayleigh distribution.

Operations on multiple independent normal variablesEdit

Any linear combination of independent normal deviates is a normal deviate.
If <math display=inline>X_1, X_2, \ldots, X_n</math> are independent standard normal random variables, then the sum of their squares has the chi-squared distribution with Template:Tmath degrees of freedom <math display=block>X_1^2 + \cdots + X_n^2 \sim \chi_n^2.</math>
If <math display=inline>X_1, X_2, \ldots, X_n</math> are independent normally distributed random variables with means Template:Tmath and variances <math display=inline>\sigma^2</math>, then their sample mean is independent from the sample standard deviation,<ref>Template:Cite journal</ref> which can be demonstrated using Basu's theorem or Cochran's theorem.<ref>Template:Cite journal</ref> The ratio of these two quantities will have the Student's t-distribution with <math display=inline>n-1</math> degrees of freedom: <math display=block>t = \frac{\overline X - \mu}{S/\sqrt{n}} = \frac{\frac{1}{n}(X_1+\cdots+X_n) - \mu}{\sqrt{\frac{1}{n(n-1)}\left[(X_1-\overline X)^2 + \cdots+(X_n-\overline X)^2\right]}} \sim t_{n-1}.</math>
If <math display=inline>X_1, X_2, \ldots, X_n</math>, <math display=inline>Y_1, Y_2, \ldots, Y_m</math> are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution with Template:Math degrees of freedom:<ref>Template:Cite book</ref> <math display=block>F = \frac{\left(X_1^2+X_2^2+\cdots+X_n^2\right)/n}{\left(Y_1^2+Y_2^2+\cdots+Y_m^2\right)/m} \sim F_{n,m}.</math>

Operations on multiple correlated normal variablesEdit

A quadratic form of a normal vector, i.e. a quadratic function <math display=inline>q = \sum x_i^2 + \sum x_j + c</math> of multiple independent or correlated normal variables, is a generalized chi-square variable.

Operations on the density functionEdit

The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution results from rescaling a section of a single density function.

Infinite divisibility and Cramér's theoremEdit

For any positive integer Template:Mvar, any normal distribution with mean Template:Tmath and variance <math display=inline>\sigma^2</math> is the distribution of the sum of Template:Mvar independent normal deviates, each with mean <math display=inline>\frac{\mu}{n}</math> and variance <math display=inline>\frac{\sigma^2}{n}</math>. This property is called infinite divisibility.<ref>Template:Harvtxt</ref>

Conversely, if <math display=inline>X_1</math> and <math display=inline>X_2</math> are independent random variables and their sum <math display=inline>X_1+X_2</math> has a normal distribution, then both <math display=inline>X_1</math> and <math display=inline>X_2</math> must be normal deviates.<ref>Template:Harvtxt</ref>

This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.<ref name="Bryc 1995 35">Template:Harvtxt</ref>

The Kac–Bernstein theoremEdit

The Kac–Bernstein theorem states that if <math display="inline">X</math> and Template:Tmath are independent and <math display=inline>X + Y</math> and <math display=inline>X - Y</math> are also independent, then both X and Y must necessarily have normal distributions.<ref name="Lukacs">Template:Harvtxt</ref><ref>Template:Cite journal</ref>

More generally, if <math display=inline>X_1, \ldots, X_n</math> are independent random variables, then two distinct linear combinations <math display=inline>\sum{a_kX_k}</math> and <math display=inline>\sum{b_kX_k}</math>will be independent if and only if all <math display=inline>X_k</math> are normal and <math display=inline>\sum{a_kb_k\sigma_k^2=0}</math>, where <math display=inline>\sigma_k^2</math> denotes the variance of <math display=inline>X_k</math>.<ref name="Lukacs" />

ExtensionsEdit

The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called normal or Gaussian laws, so a certain ambiguity in names exists.

The multivariate normal distribution describes the Gaussian law in the Template:Mvar-dimensional Euclidean space. A vector Template:Math is multivariate-normally distributed if any linear combination of its components Template:Math has a (univariate) normal distribution. The variance of Template:Mvar is a Template:Math symmetric positive-definite matrix Template:Mvar. The multivariate normal distribution is a special case of the elliptical distributions. As such, its iso-density loci in the Template:Math case are ellipses and in the case of arbitrary Template:Mvar are ellipsoids.
Rectified Gaussian distribution a rectified version of normal distribution with all the negative elements reset to 0.
Complex normal distribution deals with the complex normal vectors. A complex vector Template:Math is said to be normal if both its real and imaginary components jointly possess a Template:Math-dimensional multivariate normal distribution. The variance-covariance structure of Template:Mvar is described by two matrices: the Template:Dfn matrix Template:Math, and the Template:Dfn matrix Template:Mvar.
Matrix normal distribution describes the case of normally distributed matrices.
Gaussian processes are the normally distributed stochastic processes. These can be viewed as elements of some infinite-dimensional Hilbert space Template:Mvar, and thus are the analogues of multivariate normal vectors for the case Template:Math. A random element Template:Math is said to be normal if for any constant Template:Math the scalar product Template:Math has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear covariance operator Template:Math. Several Gaussian processes became popular enough to have their own names:
- Brownian motion;
- Brownian bridge; and
- Ornstein–Uhlenbeck process.
Gaussian q-distribution is an abstract mathematical construction that represents a q-analogue of the normal distribution.
the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. This distribution is different from the Gaussian q-distribution above.
The [[Kaniadakis Gaussian distribution|Kaniadakis Template:Mvar-Gaussian distribution]] is a generalization of the Gaussian distribution which arises from the Kaniadakis statistics, being one of the Kaniadakis distributions.

A random variable Template:Mvar has a two-piece normal distribution if it has a distribution

<math display=block>f_X( x ) = \begin{cases} N( \mu, \sigma_1^2 ),& \text{ if } x \le \mu \\ N( \mu, \sigma_2^2 ),& \text{ if } x \ge \mu \end{cases}</math>

where Template:Mvar is the mean and Template:Math and Template:Math are the variances of the distribution to the left and right of the mean respectively.

The mean Template:Math, variance Template:Math, and third central moment Template:Math of this distribution have been determined<ref name="John-1982">Template:Cite journal</ref>

<math display=block>\begin{align} \operatorname{E}( X ) &= \mu + \sqrt{\frac 2 \pi } ( \sigma_2 - \sigma_1 ), \\ \operatorname{V}( X ) &= \left( 1 - \frac 2 \pi\right)( \sigma_2 - \sigma_1 )^2 + \sigma_1 \sigma_2, \\ \operatorname{T}( X ) &= \sqrt{ \frac 2 \pi}( \sigma_2 - \sigma_1 ) \left[ \left( \frac 4 \pi - 1 \right) ( \sigma_2 - \sigma_1)^2 + \sigma_1 \sigma_2 \right]. \end{align}</math>

One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:

Pearson distribution — a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.
The generalized normal distribution, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.

Statistical inferenceEdit

Estimation of parametersEdit

Template:See also

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate them. That is, having a sample <math display=inline>(x_1, \ldots, x_n)</math> from a normal <math display=inline>\mathcal{N}(\mu, \sigma^2)</math> population we would like to learn the approximate values of parameters Template:Tmath and <math display=inline>\sigma^2</math>. The standard approach to this problem is the maximum likelihood method, which requires maximization of the log-likelihood function:Template:Anchor <math display=block>

  \ln\mathcal{L}(\mu,\sigma^2)
    = \sum_{i=1}^n \ln f(x_i\mid\mu,\sigma^2)
    = -\frac{n}{2}\ln(2\pi) - \frac{n}{2}\ln\sigma^2 - \frac{1}{2\sigma^2}\sum_{i=1}^n (x_i-\mu)^2.

</math> Taking derivatives with respect to Template:Tmath and <math display=inline>\sigma^2</math> and solving the resulting system of first order conditions yields the maximum likelihood estimates: <math display=block>

   \hat{\mu} = \overline{x} \equiv \frac{1}{n}\sum_{i=1}^n x_i, \qquad
   \hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \overline{x})^2.

</math>

Then <math display=inline>\ln\mathcal{L}(\hat{\mu},\hat{\sigma}^2)</math> is as follows:

<math display=block>\ln\mathcal{L}(\hat{\mu},\hat{\sigma}^2)

    = (-n/2) [\ln(2 \pi \hat{\sigma}^2)+1]</math>

Sample meanEdit

Template:See also

Estimator <math style="vertical-align:-.3em">\textstyle\hat\mu</math> is called the sample mean, since it is the arithmetic mean of all observations. The statistic <math style="vertical-align:0">\textstyle\overline{x}</math> is complete and sufficient for Template:Tmath, and therefore by the Lehmann–Scheffé theorem, <math style="vertical-align:-.3em">\textstyle\hat\mu</math> is the uniformly minimum variance unbiased (UMVU) estimator.<ref name="Krishnamoorthy">Template:Harvtxt</ref> In finite samples it is distributed normally: <math display=block>

   \hat\mu \sim \mathcal{N}(\mu,\sigma^2/n).

</math> The variance of this estimator is equal to the μμ-element of the inverse Fisher information matrix <math style="vertical-align:0">\textstyle\mathcal{I}^{-1}</math>. This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error of <math style="vertical-align:-.3em">\textstyle\hat\mu</math> is proportional to <math style="vertical-align:-.3em">\textstyle1/\sqrt{n}</math>, that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulations.

From the standpoint of the asymptotic theory, <math style="vertical-align:-.3em">\textstyle\hat\mu</math> is consistent, that is, it converges in probability to Template:Tmath as <math display=inline>n\rightarrow\infty</math>. The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples: <math display=block>

   \sqrt{n}(\hat\mu-\mu) \,\xrightarrow{d}\, \mathcal{N}(0,\sigma^2).

</math>

Sample varianceEdit

Template:See also

The estimator <math style="vertical-align:0">\textstyle\hat\sigma^2</math> is called the sample variance, since it is the variance of the sample (<math display=inline>(x_1, \ldots, x_n)</math>). In practice, another estimator is often used instead of the <math style="vertical-align:0">\textstyle\hat\sigma^2</math>. This other estimator is denoted <math display=inline>s^2</math>, and is also called the sample variance, which represents a certain ambiguity in terminology; its square root Template:Tmath is called the sample standard deviation. The estimator <math display=inline>s^2</math> differs from <math style="vertical-align:0">\textstyle\hat\sigma^2</math> by having Template:Math instead of n in the denominator (the so-called Bessel's correction): <math display=block>

   s^2 = \frac{n}{n-1} \hat\sigma^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2.

</math> The difference between <math display=inline>s^2</math> and <math style="vertical-align:0">\textstyle\hat\sigma^2</math> becomes negligibly small for large nTemplate:'s. In finite samples however, the motivation behind the use of <math display=inline>s^2</math> is that it is an unbiased estimator of the underlying parameter <math display=inline>\sigma^2</math>, whereas <math style="vertical-align:0">\textstyle\hat\sigma^2</math> is biased. Also, by the Lehmann–Scheffé theorem the estimator <math display=inline>s^2</math> is uniformly minimum variance unbiased (UMVU),<ref name="Krishnamoorthy" /> which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator <math style="vertical-align:0">\textstyle\hat\sigma^2</math> is better than the <math display=inline>s^2</math> in terms of the mean squared error (MSE) criterion. In finite samples both <math display=inline>s^2</math> and <math style="vertical-align:0">\textstyle\hat\sigma^2</math> have scaled chi-squared distribution with Template:Math degrees of freedom: <math display=block>

   s^2 \sim \frac{\sigma^2}{n-1} \cdot \chi^2_{n-1}, \qquad
   \hat\sigma^2 \sim \frac{\sigma^2}{n} \cdot \chi^2_{n-1}.

</math> The first of these expressions shows that the variance of <math display=inline>s^2</math> is equal to <math display=inline>2\sigma^4/(n-1)</math>, which is slightly greater than the σσ-element of the inverse Fisher information matrix <math style="vertical-align:0">\textstyle\mathcal{I}^{-1}</math>, which is <math display=inline>2\sigma^4/n</math>. Thus, <math display=inline>s^2</math> is not an efficient estimator for <math display=inline>\sigma^2</math>, and moreover, since <math display=inline>s^2</math> is UMVU, we can conclude that the finite-sample efficient estimator for <math display=inline>\sigma^2</math> does not exist.

Applying the asymptotic theory, both estimators <math display=inline>s^2</math> and <math style="vertical-align:0">\textstyle\hat\sigma^2</math> are consistent, that is they converge in probability to <math display=inline>\sigma^2</math> as the sample size <math display=inline>n\rightarrow\infty</math>. The two estimators are also both asymptotically normal: <math display=block>

   \sqrt{n}(\hat\sigma^2 - \sigma^2) \simeq
   \sqrt{n}(s^2-\sigma^2) \,\xrightarrow{d}\, \mathcal{N}(0,2\sigma^4).

</math> In particular, both estimators are asymptotically efficient for <math display=inline>\sigma^2</math>.

Confidence intervalsEdit

Template:See also

By Cochran's theorem, for normal distributions the sample mean <math style="vertical-align:-.3em">\textstyle\hat\mu</math> and the sample variance s² are independent, which means there can be no gain in considering their joint distribution. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between <math style="vertical-align:-.3em">\textstyle\hat\mu</math> and s can be employed to construct the so-called t-statistic: <math display=block>

   t = \frac{\hat\mu-\mu}{s/\sqrt{n}} = \frac{\overline{x}-\mu}{\sqrt{\frac{1}{n(n-1)}\sum(x_i-\overline{x})^2}} \sim t_{n-1}

</math> This quantity t has the Student's t-distribution with Template:Math degrees of freedom, and it is an ancillary statistic (independent of the value of the parameters). Inverting the distribution of this t-statistics will allow us to construct the confidence interval for μ;<ref>Template:Harvtxt</ref> similarly, inverting the χ² distribution of the statistic s² will give us the confidence interval for σ²:<ref>Template:Harvtxt</ref> <math display=block>\mu \in \left[ \hat\mu - t_{n-1,1-\alpha/2} \frac{s}{\sqrt{n}},\,

                                  \hat\mu + t_{n-1,1-\alpha/2} \frac{s}{\sqrt{n}} \right]</math>

<math display=block>\sigma^2 \in \left[ \frac{n-1}{\chi^2_{n-1,1-\alpha/2}}s^2,\,

                                       \frac{n-1}{\chi^2_{n-1,\alpha/2}}s^2\right]</math>

where t_k,p and Template:SubSup are the pth quantiles of the t- and χ²-distributions respectively. These confidence intervals are of the confidence level Template:Math, meaning that the true values μ and σ² fall outside of these intervals with probability (or significance level) α. In practice people usually take Template:Math, resulting in the 95% confidence intervals. The confidence interval for σ can be found by taking the square root of the interval bounds for σ².

Approximate formulas can be derived from the asymptotic distributions of <math style="vertical-align:-.3em">\textstyle\hat\mu</math> and s²: <math display=block>\mu \in

             \left[ \hat\mu - \frac{|z_{\alpha/2}|}{\sqrt n}s,\,
                     \hat\mu + \frac{|z_{\alpha/2}|}{\sqrt n}s \right]</math>

<math display=block>\sigma^2 \in

                  \left[ s^2 - \sqrt{2}\frac{|z_{\alpha/2}|}{\sqrt{n}} s^2 ,\,
                         s^2 + \sqrt{2}\frac{|z_{\alpha/2}|}{\sqrt{n}} s^2 \right]</math>

The approximate formulas become valid for large values of n, and are more convenient for the manual calculation since the standard normal quantiles z_α/2 do not depend on n. In particular, the most popular value of Template:Math, results in Template:Math.

Normality testsEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

Normality tests assess the likelihood that the given data set {x₁, ..., x_n} comes from a normal distribution. Typically the null hypothesis H₀ is that the observations are distributed normally with unspecified mean μ and variance σ², versus the alternative H_a that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

Diagnostic plots are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.

Q–Q plot, also known as normal probability plot or rankit plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ⁻¹(p_k), x_(k)), where plotting points p_k are equal to p_k = (k − α)/(n + 1 − 2α) and α is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
P–P plot – similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(z_(k)), p_k), where <math display=inline>\textstyle z_{(k)} = (x_{(k)}-\hat\mu)/\hat\sigma</math>. For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).

Goodness-of-fit tests:

Moment-based tests:

D'Agostino's K-squared test
Jarque–Bera test
Shapiro–Wilk test: This is based on the fact that the line in the Q–Q plot has the slope of σ. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.

Tests based on the empirical distribution function:

Anderson–Darling test
Lilliefors test (an adaptation of the Kolmogorov–Smirnov test)

Bayesian analysis of the normal distributionEdit

Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:

Either the mean, or the variance, or neither, may be considered a fixed quantity.
When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
Both univariate and multivariate cases need to be considered.
Either conjugate or improper prior distributions may be placed on the unknown variables.
An additional set of cases occurs in Bayesian linear regression, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficients. The resulting analysis is similar to the basic cases of independent identically distributed data.

The formulas for the non-linear-regression cases are summarized in the conjugate prior article.

Sum of two quadraticsEdit

Scalar formEdit

The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.

<math display=block>a(x-y)^2 + b(x-z)^2 = (a + b)\left(x - \frac{ay+bz}{a+b}\right)^2 + \frac{ab}{a+b}(y-z)^2</math>

This equation rewrites the sum of two quadratics in x by expanding the squares, grouping the terms in x, and completing the square. Note the following about the complex constant factors attached to some of the terms:

The factor <math display=inline>\frac{ay+bz}{a+b}</math> has the form of a weighted average of y and z.
<math display=inline>\frac{ab}{a+b} = \frac{1}{\frac{1}{a}+\frac{1}{b}} = (a^{-1} + b^{-1})^{-1}.</math> This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities a and b add directly, so to combine a and b themselves, it is necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean, so it is not surprising that <math display=inline>\frac{ab}{a+b}</math> is one-half the harmonic mean of a and b.

Vector formEdit

A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length k, and A and B are symmetric, invertible matrices of size <math display=inline>k\times k</math>, then

<math display=block> \begin{align} & (\mathbf{y}-\mathbf{x})'\mathbf{A}(\mathbf{y}-\mathbf{x}) + (\mathbf{x}-\mathbf{z})' \mathbf{B}(\mathbf{x}-\mathbf{z}) \\ = {} & (\mathbf{x} - \mathbf{c})'(\mathbf{A}+\mathbf{B})(\mathbf{x} - \mathbf{c}) + (\mathbf{y} - \mathbf{z})'(\mathbf{A}^{-1} + \mathbf{B}^{-1})^{-1}(\mathbf{y} - \mathbf{z}) \end{align} </math>

where

<math display=block>\mathbf{c} = (\mathbf{A} + \mathbf{B})^{-1}(\mathbf{A}\mathbf{y} + \mathbf{B} \mathbf{z})</math>

The form x′ A x is called a quadratic form and is a scalar: <math display=block>\mathbf{x}'\mathbf{A}\mathbf{x} = \sum_{i,j}a_{ij} x_i x_j</math> In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since <math display=inline>x_i x_j = x_j x_i</math>, only the sum <math display=inline>a_{ij} + a_{ji}</math> matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric. Furthermore, if A is symmetric, then the form <math display=inline>\mathbf{x}'\mathbf{A}\mathbf{y} = \mathbf{y}'\mathbf{A}\mathbf{x}.</math>

Sum of differences from the meanEdit

Another useful formula is as follows: <math display=block>\sum_{i=1}^n (x_i-\mu)^2 = \sum_{i=1}^n (x_i-\bar{x})^2 + n(\bar{x} -\mu)^2</math> where <math display=inline>\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i.</math>

With known varianceEdit

For a set of i.i.d. normally distributed data points X of size n where each individual point x follows <math display=inline>x \sim \mathcal{N}(\mu, \sigma^2)</math> with known variance σ², the conjugate prior distribution is also normally distributed.

This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ². Then if <math display=inline>x \sim \mathcal{N}(\mu, 1/\tau)</math> and <math display=inline>\mu \sim \mathcal{N}(\mu_0, 1/\tau_0),</math> we proceed as follows.

First, the likelihood function is (using the formula above for the sum of differences from the mean):

<math display=block>\begin{align} p(\mathbf{X}\mid\mu,\tau) &= \prod_{i=1}^n \sqrt{\frac{\tau}{2\pi}} \exp\left(-\frac{1}{2}\tau(x_i-\mu)^2\right) \\ &= \left(\frac{\tau}{2\pi}\right)^{n/2} \exp\left(-\frac{1}{2}\tau \sum_{i=1}^n (x_i-\mu)^2\right) \\ &= \left(\frac{\tau}{2\pi}\right)^{n/2} \exp\left[-\frac{1}{2}\tau \left(\sum_{i=1}^n(x_i-\bar{x})^2 + n(\bar{x} -\mu)^2\right)\right]. \end{align}</math>

Then, we proceed as follows:

<math display=block>\begin{align} p(\mu\mid\mathbf{X}) &\propto p(\mathbf{X}\mid\mu) p(\mu) \\ & = \left(\frac{\tau}{2\pi}\right)^{n/2} \exp\left[-\frac{1}{2}\tau \left(\sum_{i=1}^n(x_i-\bar{x})^2 + n(\bar{x} -\mu)^2\right)\right] \sqrt{\frac{\tau_0}{2\pi}} \exp\left(-\frac{1}{2}\tau_0(\mu-\mu_0)^2\right) \\ &\propto \exp\left(-\frac{1}{2}\left(\tau\left(\sum_{i=1}^n(x_i-\bar{x})^2 + n(\bar{x} -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\ &\propto \exp\left(-\frac{1}{2} \left(n\tau(\bar{x}-\mu)^2 + \tau_0(\mu-\mu_0)^2 \right)\right) \\ &= \exp\left(-\frac{1}{2}(n\tau + \tau_0)\left(\mu - \dfrac{n\tau \bar{x} + \tau_0\mu_0}{n\tau + \tau_0}\right)^2 + \frac{n\tau\tau_0}{n\tau+\tau_0}(\bar{x} - \mu_0)^2\right) \\ &\propto \exp\left(-\frac{1}{2}(n\tau + \tau_0)\left(\mu - \dfrac{n\tau \bar{x} + \tau_0\mu_0}{n\tau + \tau_0}\right)^2\right) \end{align}</math>

In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving μ. The result is the kernel of a normal distribution, with mean <math display=inline>\frac{n\tau \bar{x} + \tau_0\mu_0}{n\tau + \tau_0}</math> and precision <math display=inline>n\tau + \tau_0</math>, i.e.

<math display=block>p(\mu\mid\mathbf{X}) \sim \mathcal{N}\left(\frac{n\tau \bar{x} + \tau_0\mu_0}{n\tau + \tau_0}, \frac{1}{n\tau + \tau_0}\right)</math>

This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:

<math display=block>\begin{align} \tau_0' &= \tau_0 + n\tau \\[5pt] \mu_0' &= \frac{n\tau \bar{x} + \tau_0\mu_0}{n\tau + \tau_0} \\[5pt] \bar{x} &= \frac{1}{n}\sum_{i=1}^n x_i \end{align}</math>

That is, to combine n data points with total precision of nτ (or equivalently, total variance of n/σ²) and mean of values <math display=inline>\bar{x}</math>, derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a precision-weighted average, i.e. a weighted average of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)

The above formula reveals why it is more convenient to do Bayesian analysis of conjugate priors for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas

<math display=block>\begin{align} {\sigma^2_0}' &= \frac{1}{\frac{n}{\sigma^2} + \frac{1}{\sigma_0^2}} \\[5pt] \mu_0' &= \frac{\frac{n\bar{x}}{\sigma^2} + \frac{\mu_0}{\sigma_0^2}}{\frac{n}{\sigma^2} + \frac{1}{\sigma_0^2}} \\[5pt] \bar{x} &= \frac{1}{n}\sum_{i=1}^n x_i \end{align}</math>

With known meanEdit

For a set of i.i.d. normally distributed data points X of size n where each individual point x follows <math display=inline>x \sim \mathcal{N}(\mu, \sigma^2)</math> with known mean μ, the conjugate prior of the variance has an inverse gamma distribution or a scaled inverse chi-squared distribution. The two are equivalent except for having different parameterizations. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ² is as follows:

<math display=block>p(\sigma^2\mid\nu_0,\sigma_0^2) = \frac{(\sigma_0^2\frac{\nu_0}{2})^{\nu_0/2}}{\Gamma\left(\frac{\nu_0}{2} \right)}~\frac{\exp\left[ \frac{-\nu_0 \sigma_0^2}{2 \sigma^2}\right]}{(\sigma^2)^{1+\frac{\nu_0}{2}}} \propto \frac{\exp\left[ \frac{-\nu_0 \sigma_0^2}{2 \sigma^2}\right]}{(\sigma^2)^{1+\frac{\nu_0}{2}}}</math>

The likelihood function from above, written in terms of the variance, is:

<math display=block>\begin{align} p(\mathbf{X}\mid\mu,\sigma^2) &= \left(\frac{1}{2\pi\sigma^2}\right)^{n/2} \exp\left[-\frac{1}{2\sigma^2} \sum_{i=1}^n (x_i-\mu)^2\right] \\ &= \left(\frac{1}{2\pi\sigma^2}\right)^{n/2} \exp\left[-\frac{S}{2\sigma^2}\right] \end{align}</math>

where

Then:

<math display=block>\begin{align} p(\sigma^2\mid\mathbf{X}) &\propto p(\mathbf{X}\mid\sigma^2) p(\sigma^2) \\ &= \left(\frac{1}{2\pi\sigma^2}\right)^{n/2} \exp\left[-\frac{S}{2\sigma^2}\right] \frac{(\sigma_0^2\frac{\nu_0}{2})^{\frac{\nu_0}{2}}}{\Gamma\left(\frac{\nu_0}{2} \right)}~\frac{\exp\left[ \frac{-\nu_0 \sigma_0^2}{2 \sigma^2}\right]}{(\sigma^2)^{1+\frac{\nu_0}{2}}} \\ &\propto \left(\frac{1}{\sigma^2}\right)^{n/2} \frac{1}{(\sigma^2)^{1+\frac{\nu_0}{2}}} \exp\left[-\frac{S}{2\sigma^2} + \frac{-\nu_0 \sigma_0^2}{2 \sigma^2}\right] \\ &= \frac{1}{(\sigma^2)^{1+\frac{\nu_0+n}{2}}} \exp\left[-\frac{\nu_0 \sigma_0^2 + S}{2\sigma^2}\right] \end{align}</math>

The above is also a scaled inverse chi-squared distribution where

<math display=block>\begin{align} \nu_0' &= \nu_0 + n \\ \nu_0'{\sigma_0^2}' &= \nu_0 \sigma_0^2 + \sum_{i=1}^n (x_i-\mu)^2 \end{align}</math>

or equivalently

<math display=block>\begin{align} \nu_0' &= \nu_0 + n \\ {\sigma_0^2}' &= \frac{\nu_0 \sigma_0^2 + \sum_{i=1}^n (x_i-\mu)^2}{\nu_0+n} \end{align}</math>

Reparameterizing in terms of an inverse gamma distribution, the result is:

<math display=block>\begin{align} \alpha' &= \alpha + \frac{n}{2} \\ \beta' &= \beta + \frac{\sum_{i=1}^n (x_i-\mu)^2}{2} \end{align}</math>

With unknown mean and unknown varianceEdit

For a set of i.i.d. normally distributed data points X of size n where each individual point x follows <math display=inline>x \sim \mathcal{N}(\mu, \sigma^2)</math> with unknown mean μ and unknown variance σ², a combined (multivariate) conjugate prior is placed over the mean and variance, consisting of a normal-inverse-gamma distribution. Logically, this originates as follows:

From the analysis of the case with unknown mean but known variance, we see that the update equations involve sufficient statistics computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.
From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and sum of squared deviations.
Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.
To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.
This suggests that we create a conditional prior of the mean on the unknown variance, with a hyperparameter specifying the mean of the pseudo-observations associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.
This leads immediately to the normal-inverse-gamma distribution, which is the product of the two distributions just defined, with conjugate priors used (an inverse gamma distribution over the variance, and a normal distribution over the mean, conditional on the variance) and with the same four parameters just defined.

The priors are normally defined as follows:

<math display=block>\begin{align} p(\mu\mid\sigma^2; \mu_0, n_0) &\sim \mathcal{N}(\mu_0,\sigma^2/n_0) \\ p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2) \end{align}</math>

The update equations can be derived, and look as follows:

<math display=block>\begin{align} \bar{x} &= \frac 1 n \sum_{i=1}^n x_i \\ \mu_0' &= \frac{n_0\mu_0 + n\bar{x}}{n_0 + n} \\ n_0' &= n_0 + n \\ \nu_0' &= \nu_0 + n \\ \nu_0'{\sigma_0^2}' &= \nu_0 \sigma_0^2 + \sum_{i=1}^n (x_i-\bar{x})^2 + \frac{n_0 n}{n_0 + n}(\mu_0 - \bar{x})^2 \end{align}</math>

The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for <math display=inline>\nu_0'{\sigma_0^2}'</math> is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.

Template:Math proof} \exp\left(-\frac{n_0}{2\sigma^2}(\mu-\mu_0)^2\right) \\ &\propto (\sigma^2)^{-1/2} \exp\left(-\frac{n_0}{2\sigma^2}(\mu-\mu_0)^2\right) \\ p(\sigma^2; \nu_0,\sigma_0^2) &\sim I\chi^2(\nu_0,\sigma_0^2) = IG(\nu_0/2, \nu_0\sigma_0^2/2) \\ &= \frac{(\sigma_0^2\nu_0/2)^{\nu_0/2}}{\Gamma(\nu_0/2)}~\frac{\exp\left[ \frac{-\nu_0 \sigma_0^2}{2 \sigma^2}\right]}{(\sigma^2)^{1+\nu_0/2}} \\ &\propto {(\sigma^2)^{-(1+\nu_0/2)}} \exp\left[ \frac{-\nu_0 \sigma_0^2}{2 \sigma^2}\right]. \end{align}</math>

Therefore, the joint prior is

<math display=block>\begin{align} p(\mu,\sigma^2; \mu_0, n_0, \nu_0,\sigma_0^2) &= p(\mu\mid\sigma^2; \mu_0, n_0)\,p(\sigma^2; \nu_0,\sigma_0^2) \\ &\propto (\sigma^2)^{-(\nu_0+3)/2} \exp\left[-\frac 1 {2\sigma^2}\left(\nu_0\sigma_0^2 + n_0(\mu-\mu_0)^2\right)\right]. \end{align}</math>

The likelihood function from the section above with known variance is:

<math display=block>\begin{align} p(\mathbf{X}\mid\mu,\sigma^2) &= \left(\frac{1}{2\pi\sigma^2}\right)^{n/2} \exp\left[-\frac{1}{2\sigma^2} \left(\sum_{i=1}^n(x_i -\mu)^2\right)\right] \end{align}</math>

Writing it in terms of variance rather than precision, we get: <math display=block>\begin{align} p(\mathbf{X}\mid\mu,\sigma^2) &= \left(\frac{1}{2\pi\sigma^2}\right)^{n/2} \exp\left[-\frac{1}{2\sigma^2} \left(\sum_{i=1}^n(x_i-\bar{x})^2 + n(\bar{x} -\mu)^2\right)\right] \\ &\propto {\sigma^2}^{-n/2} \exp\left[-\frac{1}{2\sigma^2} \left(S + n(\bar{x} -\mu)^2\right)\right] \end{align}</math> where <math display=inline>S = \sum_{i=1}^n(x_i-\bar{x})^2.</math>

Therefore, the posterior is (dropping the hyperparameters as conditioning factors): <math display=block>\begin{align} p(\mu,\sigma^2\mid\mathbf{X}) & \propto p(\mu,\sigma^2) \, p(\mathbf{X}\mid\mu,\sigma^2) \\ & \propto (\sigma^2)^{-(\nu_0+3)/2} \exp\left[-\frac{1}{2\sigma^2}\left(\nu_0\sigma_0^2 + n_0(\mu-\mu_0)^2\right)\right] {\sigma^2}^{-n/2} \exp\left[-\frac{1}{2\sigma^2} \left(S + n(\bar{x} -\mu)^2\right)\right] \\ &= (\sigma^2)^{-(\nu_0+n+3)/2} \exp\left[-\frac{1}{2\sigma^2}\left(\nu_0\sigma_0^2 + S + n_0(\mu-\mu_0)^2 + n(\bar{x} -\mu)^2\right)\right] \\ &= (\sigma^2)^{-(\nu_0+n+3)/2} \exp\left[-\frac{1}{2\sigma^2}\left(\nu_0\sigma_0^2 + S + \frac{n_0 n}{n_0+n}(\mu_0-\bar{x})^2 + (n_0+n)\left(\mu-\frac{n_0\mu_0 + n\bar{x}}{n_0 + n}\right)^2\right)\right] \\ & \propto (\sigma^2)^{-1/2} \exp\left[-\frac{n_0+n}{2\sigma^2}\left(\mu-\frac{n_0\mu_0 + n\bar{x}}{n_0 + n}\right)^2\right] \\ & \quad\times (\sigma^2)^{-(\nu_0/2+n/2+1)} \exp\left[-\frac{1}{2\sigma^2}\left(\nu_0\sigma_0^2 + S + \frac{n_0 n}{n_0+n}(\mu_0-\bar{x})^2\right)\right] \\ & = \mathcal{N}_{\mu\mid\sigma^2}\left(\frac{n_0\mu_0 + n\bar{x}}{n_0 + n}, \frac{\sigma^2}{n_0+n}\right) \cdot {\rm IG}_{\sigma^2}\left(\frac12(\nu_0+n), \frac12\left(\nu_0\sigma_0^2 + S + \frac{n_0 n}{n_0+n}(\mu_0-\bar{x})^2\right)\right). \end{align}</math>

In other words, the posterior distribution has the form of a product of a normal distribution over <math display=inline>p(\mu|\sigma^2)</math> times an inverse gamma distribution over <math display=inline>p(\sigma^2)</math>, with parameters that are the same as the update equations above. }}

Occurrence and applicationsEdit

The occurrence of normal distribution in practical problems can be loosely classified into four categories:

Exactly normal distributions;
Approximately normal laws, for example when such approximation is justified by the central limit theorem; and
Distributions modeled as normal – the normal distribution being the distribution with maximum entropy for a given mean and variance.
Regression problems – the normal distribution being found after systematic effects have been modeled sufficiently well.

Exact normalityEdit

File:QHarmonicOscillator.png

The ground state of a quantum harmonic oscillator has the Gaussian distribution.

A normal distribution occurs in some physical theories:

The velocity distribution of independently moving and perfectly elastic spheres, which is a consequence of Maxwell's Dynamical Theory of Gases, Part I (1860).Template:Sfnp
The ground state wave function in position space of the quantum harmonic oscillator.<ref>Template:Cite book</ref>
The position of a particle that experiences diffusion. If initially the particle is located at a specific point (that is its probability distribution is the Dirac delta function), then after time t its location is described by a normal distribution with variance t, which satisfies the diffusion equation <math display=inline>\frac{\partial}{\partial t} f(x,t) = \frac{1}{2} \frac{\partial^2}{\partial x^2} f(x,t)</math>. If the initial location is given by a certain density function <math display=inline>g(x)</math>, then the density at time t is the convolution of g and the normal probability density function.

Approximate normalityEdit

Approximately normal distributions occur in many situations, as explained by the central limit theorem. When the outcome is produced by many small effects acting additively and independently, its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.

In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible and decomposable distributions are involved, such as
- Binomial random variables, associated with binary response variables;
- Poisson random variables, associated with rare events;
Thermal radiation has a Bose–Einstein distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.

Assumed normalityEdit

File:Fisher iris versicolor sepalwidth.svg

Histogram of sepal widths for Iris versicolor from Fisher's Iris flower data set, with superimposed best-fitting normal distribution

I can only recognize the occurrence of the normal curve – the Laplacian curve of errors – as a very abnormal phenomenon. It is roughly approximated to in certain distributions; for this reason, and on account for its beautiful simplicity, we may, perhaps, use it as a first approximation, particularly in theoretical investigations.{{#if:Template:Harvtxt|{{#if:|}}
— {{#if:|, in }}Template:Comma separated entries
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There are statistical methods to empirically test that assumption; see the above Normality tests section.

In biology, the logarithm of various variables tend to have a normal distribution, that is, they tend to have a log-normal distribution (after separation on male/female subpopulations), with examples including:
- Measures of size of living tissue (length, height, skin area, weight);<ref>Template:Harvtxt</ref>
- The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth; presumably the thickness of tree bark also falls under this category;
- Certain physiological measurements, such as blood pressure of adult humans.
In finance, in particular the Black–Scholes model, changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest, not like simple interest, and so are multiplicative). Some mathematicians such as Benoit Mandelbrot have argued that log-Levy distributions, which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crashes. The use of the assumption of normal distribution occurring in financial models has also been criticized by Nassim Nicholas Taleb in his works.
Measurement errors in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.<ref>Template:Cite book</ref>
In standardized testing, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the IQ test) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the SAT's traditional range of 200–800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.

File:FitNormDistr.tif

Fitted cumulative normal distribution to October rainfalls, see distribution fitting

Many scores are derived from the normal distribution, including percentile ranks (percentiles or quantiles), normal curve equivalents, stanines, z-scores, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, t-tests and ANOVAs. Bell curve grading assigns relative grades based on a normal distribution of scores.
In hydrology the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the central limit theorem.<ref>Template:Cite book</ref> The blue picture, made with CumFreq, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Methodological problems and peer reviewEdit

John Ioannidis argued that using normally distributed standard deviations as standards for validating research findings leave falsifiable predictions about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.<ref>Why Most Published Research Findings Are False, John P. A. Ioannidis, 2005</ref>

Computational methodsEdit

Generating values from normal distributionEdit

File:Planche de Galton.jpg

The bean machine, a device invented by Francis Galton, can be called the first generator of normal random variables. This machine consists of a vertical board with interleaved rows of pins. Small balls are dropped from the top and then bounce randomly left or right as they hit the pins. The balls are collected into bins at the bottom and settle down into a pattern resembling the Gaussian curve.

In computer simulations, especially in applications of the Monte-Carlo method, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a Template:Math can be generated as Template:Math, where Z is standard normal. All these algorithms rely on the availability of a random number generator U capable of producing uniform random variates.

The most straightforward method is based on the probability integral transform property: if U is distributed uniformly on (0,1), then Φ⁻¹(U) will have the standard normal distribution. The drawback of this method is that it relies on calculation of the probit function Φ⁻¹, which cannot be done analytically. Some approximate methods are described in Template:Harvtxt and in the erf article. Wichura gives a fast algorithm for computing this function to 16 decimal places,<ref>Template:Cite journal</ref> which is used by R to compute random variates of the normal distribution.
An easy-to-program approximate approach that relies on the central limit theorem is as follows: generate 12 uniform U(0,1) deviates, add them all up, and subtract 6 – the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be Irwin–Hall, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (−6, 6).<ref>Template:Harvtxt</ref> Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.
The Box–Muller method uses two independent random numbers U and V distributed uniformly on (0,1). Then the two random variables X and Y <math display=block>

   X = \sqrt{- 2 \ln U} \, \cos(2 \pi V) , \qquad
   Y = \sqrt{- 2 \ln U} \, \sin(2 \pi V) .

</math> will both have the standard normal distribution, and will be independent. This formulation arises because for a bivariate normal random vector (X, Y) the squared norm Template:Math will have the chi-squared distribution with two degrees of freedom, which is an easily generated exponential random variable corresponding to the quantity −2 ln(U) in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable V.

The Marsaglia polar method is a modification of the Box–Muller method which does not require computation of the sine and cosine functions. In this method, U and V are drawn from the uniform (−1,1) distribution, and then Template:Math is computed. If S is greater or equal to 1, then the method starts over, otherwise the two quantities <math display=block>X = U\sqrt{\frac{-2\ln S}{S}}, \qquad Y = V\sqrt{\frac{-2\ln S}{S}}</math> are returned. Again, X and Y are independent, standard normal random variables.
The Ratio method<ref>Template:Harvtxt</ref> is a rejection method. The algorithm proceeds as follows:
- Generate two independent uniform deviates U and V;
- Compute X = Template:Sqrt (V − 0.5)/U;
- Optional: if X² ≤ 5 − 4e^1/4U then accept X and terminate algorithm;
- Optional: if X² ≥ 4e^−1.35/U + 1.4 then reject X and start over from step 1;
- If X² ≤ −4 lnU then accept X, otherwise start over the algorithm.
The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved<ref>Template:Harvtxt</ref> so that the logarithm is rarely evaluated.
The ziggurat algorithm<ref>Template:Harvtxt</ref> is faster than the Box–Muller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.
Integer arithmetic can be used to sample from the standard normal distribution.<ref>Template:Harvtxt</ref><ref>Template:Harvtxt</ref> This method is exact in the sense that it satisfies the conditions of ideal approximation;<ref>Template:Harvtxt</ref> i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.
There is also some investigation<ref>Template:Harvtxt</ref> into the connection between the fast Hadamard transform and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.

Numerical approximations for the normal cumulative distribution function and normal quantile functionEdit

The standard normal cumulative distribution function is widely used in scientific and statistical computing.

The values Φ(x) may be approximated very accurately by a variety of methods, such as numerical integration, Taylor series, asymptotic series and continued fractions. Different approximations are used depending on the desired level of accuracy.

Template:Harvtxt give the approximation for Φ(x) for x > 0 with the absolute error Template:Math (algorithm 26.2.17): <math display=block>

   \Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac{1}{1+b_0x},

</math> where ϕ(x) is the standard normal probability density function, and b₀ = 0.2316419, b₁ = 0.319381530, b₂ = −0.356563782, b₃ = 1.781477937, b₄ = −1.821255978, b₅ = 1.330274429.

Template:Harvtxt lists some dozens of approximations – by means of rational functions, with or without exponentials – for the Template:Mono function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by Template:Harvtxt combines Hart's algorithm 5666 with a continued fraction approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
Template:Harvtxt after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via Rational Chebyshev Approximation.
Template:Harvtxt suggested a simple algorithmTemplate:NoteTag based on the Taylor series expansion <math display=block>

   \Phi(x) = \frac12 + \varphi(x)\left( x + \frac{x^3} 3 + \frac{x^5}{3 \cdot 5} + \frac{x^7}{3 \cdot 5 \cdot 7} + \frac{x^9}{3 \cdot 5 \cdot 7 \cdot 9} + \cdots \right)

</math> for calculating Template:Math with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when Template:Math).

The GNU Scientific Library calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with Chebyshev polynomials.
Template:Harvtxt proposes the following approximation of <math display=inline>1-\Phi</math> with a maximum relative error less than <math display=inline>2^{-53}</math> <math display=inline>

\left(\approx 1.1 \times 10^{-16}\right) </math> in absolute value: for <math display=inline>x \ge 0</math><math display=inline> \begin{aligned}

   1-\Phi\left(x\right) & = \left(\frac{0.39894228040143268}{x+2.92678600515804815}\right)
   \left(\frac{x^2+8.42742300458043240 x+18.38871225773938487}{x^2+5.81582518933527391 x+8.97280659046817350} \right) \\
   & \left(\frac{x^2+7.30756258553673541 x+18.25323235347346525}{x^2+5.70347935898051437 x+10.27157061171363079}\right)
   \left(\frac{x^2+5.66479518878470765 x+18.61193318971775795}{x^2+5.51862483025707963 x+12.72323261907760928}\right) \\
   & \left( \frac{x^2+4.91396098895240075 x+24.14804072812762821}{x^2+5.26184239579604207 x+16.88639562007936908}\right)
   \left( \frac{x^2+3.83362947800146179 x+11.61511226260603247}{x^2+4.92081346632882033 x+24.12333774572479110}\right) e^{-\frac{x^2}{2}}

\end{aligned} </math> and for <math display=inline>

x<0

</math>, <math display=block>

   1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)

</math>

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting Template:Math, the simplest approximation for the quantile function is: <math display=block>z = \Phi^{-1}(p)=5.5556\left[1- \left( \frac{1-p} p \right)^{0.1186}\right],\qquad p\ge 1/2</math>

This approximation delivers for z a maximum absolute error of 0.026 (for Template:Math, corresponding to Template:Math). For Template:Math replace p by Template:Math and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation: <math display=block> z=-0.4115\left\{ \frac{1-p} p + \log \left[ \frac{1-p} p \right] - 1 \right\}, \qquad p\ge 1/2</math>

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by <math display=block>\begin{align} L(z) & =\int_z^\infty (u-z)\varphi(u) \, du=\int_z^\infty [1-\Phi (u)] \, du \\[5pt] L(z) & \approx \begin{cases}

  0.4115\left(\dfrac p {1-p} \right) - z, & p<1/2, \\ \\
  0.4115\left( \dfrac {1-p} p \right), & p\ge 1/2.

\end{cases} \\[5pt] \text{or, equivalently,} \\ L(z) & \approx \begin{cases}

  0.4115\left\{ 1-\log \left[ \frac p {1-p} \right] \right\}, & p < 1/2, \\ \\
  0.4115 \dfrac{1-p} p, & p\ge 1/2.

\end{cases} \end{align}</math>

This approximation is particularly accurate for the right far-tail (maximum error of 10⁻³ for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on Response Modeling Methodology (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: Error function#Approximation with elementary functions. In particular, small relative error on the whole domain for the cumulative distribution function Template:Tmath and the quantile function <math display=inline>\Phi^{-1}</math> as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.

HistoryEdit

DevelopmentEdit

Some authors<ref>Template:Harvtxt</ref><ref>Template:Harvtxt</ref> attribute the discovery of the normal distribution to de Moivre, who in 1738Template:NoteTag published in the second edition of his The Doctrine of Chances the study of the coefficients in the binomial expansion of Template:Math. De Moivre proved that the middle term in this expansion has the approximate magnitude of <math display=inline>2^n/\sqrt{2\pi n}</math>, and that "If m or Template:Sfracn be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term distant from the middle by the Interval ℓ, has to the middle Term, is <math display=inline>-\frac{2\ell\ell}{n}</math>."<ref>De Moivre, Abraham (1733), Corollary I – see Template:Harvtxt</ref> Although this theorem can be interpreted as the first obscure expression for the normal probability law, Stigler points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.<ref>Template:Harvtxt</ref>

File:Carl Friedrich Gauss.jpg

Carl Friedrich Gauss discovered the normal distribution in 1809 as a way to rationalize the method of least squares.

In 1823 Gauss published his monograph "Theoria combinationis observationum erroribus minimis obnoxiae" where among other things he introduces several important statistical concepts, such as the method of least squares, the method of maximum likelihood, and the normal distribution. Gauss used M, Template:Nobr, Template:Nobr to denote the measurements of some unknown quantity V, and sought the most probable estimator of that quantity: the one that maximizes the probability Template:Math of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function φ is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values.Template:NoteTag Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:<ref>Template:Harvtxt</ref> <math display=block>

   \varphi\mathit{\Delta} = \frac h {\surd\pi} \, e^{-\mathrm{hh}\Delta\Delta},

</math> where h is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the non-linear weighted least squares method.<ref>Template:Harvtxt</ref>

File:Pierre-Simon Laplace.jpg

Pierre-Simon Laplace proved the central limit theorem in 1810, consolidating the importance of the normal distribution in statistics.

Although Gauss was the first to suggest the normal distribution law, Laplace made significant contributions.Template:NoteTag It was Laplace who first posed the problem of aggregating several observations in 1774,<ref>Template:Harvtxt</ref> although his own solution led to the Laplacian distribution. It was Laplace who first calculated the value of the [[Gaussian integral|integral Template:Math]] in 1782, providing the normalization constant for the normal distribution.<ref>Template:Harvtxt</ref> For this accomplishment, Gauss acknowledged the priority of Laplace.<ref>Template:Harvtxt</ref> Finally, it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem, which emphasized the theoretical importance of the normal distribution.<ref>Template:Harvtxt</ref>

It is of interest to note that in 1809 an Irish-American mathematician Robert Adrain published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss.<ref>Template:Harvtxt</ref> His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by Abbe.<ref>Template:Harvtxt</ref>

In the middle of the 19th century Maxwell demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena:Template:Sfnp The number of particles whose velocity, resolved in a certain direction, lies between x and x + dx is <math display=block>

   \operatorname{N} \frac{1}{\alpha\;\sqrt\pi}\; e^{-\frac{x^2}{\alpha^2}} \, dx

</math>

NamingEdit

Today, the concept is usually known in English as the normal distribution or Gaussian distribution. Other less common names include Gauss distribution, Laplace–Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.

Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual.<ref>Jaynes, Edwin J.; Probability Theory: The Logic of Science, Ch. 7.</ref> However, by the end of the 19th century some authorsTemplate:NoteTag had started using the name normal distribution, where the word "normal" was used as an adjective – the term now being seen as a reflection of the fact that this distribution was seen as typical, common – and thus normal. Peirce (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what would, in the long run, occur under certain circumstances."<ref>Peirce, Charles S. (c. 1909 MS), Collected Papers v. 6, paragraph 327.</ref> Around the turn of the 20th century Pearson popularized the term normal as a designation for this distribution.<ref>Template:Harvtxt.</ref>

Many years ago I called the Laplace–Gaussian curve the normal curve, which name, while it avoids an international question of priority, has the disadvantage of leading people to believe that all other distributions of frequency are in one sense or another 'abnormal'. {{#if:Template:Harvtxt|{{#if:|}}
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Also, it was Pearson who first wrote the distribution in terms of the standard deviation σ as in modern notation. Soon after this, in year 1915, Fisher added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays: <math display=block> df = \frac{1}{\sqrt{2\sigma^2\pi}} e^{-(x - m)^2/(2\sigma^2)} \, dx.</math>

The term standard normal distribution, which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) Introduction to Mathematical Statistics and A. M. Mood (1950) Introduction to the Theory of Statistics.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Harvtxt introduces the terms standard normal curve (p. 33) and standard normal distribution (p. 69).</ref><ref>Template:Harvtxt explicitly defines the standard normal distribution (p. 112).</ref>

NotesEdit

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CitationsEdit

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|CitationClass=web }} In particular, the entries for "bell-shaped and bell curve", "normal (distribution)", "Gaussian", and "Error, law of error, theory of errors, etc.".

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Normal distribution

DefinitionsEdit

Standard normal distributionEdit

General normal distributionEdit

NotationEdit

Alternative parameterizationsEdit

Cumulative distribution functionEdit

Error functionEdit

Recursive computation with Taylor series expansionEdit

Using the Taylor series and Newton's method for the inverse functionEdit

Standard deviation and coverageEdit

Quantile functionEdit

PropertiesEdit

Symmetries and derivativesEdit

MomentsEdit

Fourier transform and characteristic functionEdit

Moment- and cumulant-generating functionsEdit

Stein operator and classEdit

Zero-variance limitEdit

Maximum entropyEdit

Other propertiesEdit

Related distributionsEdit

Central limit theoremEdit

Operations and functions of normal variablesEdit

Operations on a single normal variableEdit

Operations on two independent normal variablesEdit

Operations on two independent standard normal variablesEdit

Operations on multiple independent normal variablesEdit

Operations on multiple correlated normal variablesEdit

Operations on the density functionEdit

Infinite divisibility and Cramér's theoremEdit

The Kac–Bernstein theoremEdit

ExtensionsEdit

Statistical inferenceEdit

Estimation of parametersEdit

Sample meanEdit

Sample varianceEdit

Confidence intervalsEdit

Normality testsEdit

Bayesian analysis of the normal distributionEdit

Sum of two quadraticsEdit

Scalar formEdit

Vector formEdit

Sum of differences from the meanEdit

With known varianceEdit

With known meanEdit

With unknown mean and unknown varianceEdit

Occurrence and applicationsEdit

Exact normalityEdit

Approximate normalityEdit

Assumed normalityEdit

Methodological problems and peer reviewEdit

Computational methodsEdit

Generating values from normal distributionEdit

Numerical approximations for the normal cumulative distribution function and normal quantile functionEdit

HistoryEdit

DevelopmentEdit

NamingEdit

See alsoEdit

NotesEdit

ReferencesEdit

CitationsEdit

SourcesEdit

External linksEdit