Probability density function

File:Boxplot vs PDF.svg

Box plot and probability density function of a normal distribution Template:Math.

File:Visualisation mode median mean.svg

CitationClass=web }}</ref>

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would be equal to that sample.<ref>Template:Cite book</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Probability density is the probability per unit length, in other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample.

More precisely, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. This probability is given by the integral of this variable's PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and the area under the entire curve is equal to 1.

The terms probability distribution function and probability function have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values or it may refer to the cumulative distribution function, or it may be a probability mass function (PMF) rather than the density. "Density function" itself is also used for the probability mass function, leading to further confusion.<ref>Ord, J.K. (1972) Families of Frequency Distributions, Griffin. Template:Isbn (for example, Table 5.1 and Example 5.4)</ref> In general though, the PMF is used in the context of discrete random variables (random variables that take values on a countable set), while the PDF is used in the context of continuous random variables.

Example

File:4 continuous probability density functions.png

Examples of four continuous probability density functions.

Suppose bacteria of a certain species typically live 20 to 30 hours. The probability that a bacterium lives Template:Em 5 hours is equal to zero. A lot of bacteria live for approximately 5 hours, but there is no chance that any given bacterium dies at exactly 5.00... hours. However, the probability that the bacterium dies between 5 hours and 5.01 hours is quantifiable. Suppose the answer is 0.02 (i.e., 2%). Then, the probability that the bacterium dies between 5 hours and 5.001 hours should be about 0.002, since this time interval is one-tenth as long as the previous. The probability that the bacterium dies between 5 hours and 5.0001 hours should be about 0.0002, and so on.

In this example, the ratio (probability of living during an interval) / (duration of the interval) is approximately constant, and equal to 2 per hour (or 2 hour⁻¹). For example, there is 0.02 probability of dying in the 0.01-hour interval between 5 and 5.01 hours, and (0.02 probability / 0.01 hours) = 2 hour⁻¹. This quantity 2 hour⁻¹ is called the probability density for dying at around 5 hours. Therefore, the probability that the bacterium dies at 5 hours can be written as (2 hour⁻¹) dt. This is the probability that the bacterium dies within an infinitesimal window of time around 5 hours, where dt is the duration of this window. For example, the probability that it lives longer than 5 hours, but shorter than (5 hours + 1 nanosecond), is (2 hour⁻¹)×(1 nanosecond) ≈ Template:Val (using the unit conversion Template:Val nanoseconds = 1 hour).

There is a probability density function f with f(5 hours) = 2 hour⁻¹. The integral of f over any window of time (not only infinitesimal windows but also large windows) is the probability that the bacterium dies in that window.

Absolutely continuous univariate distributions

A probability density function is most commonly associated with absolutely continuous univariate distributions. A random variable <math>X</math> has density <math>f_X</math>, where <math>f_X</math> is a non-negative Lebesgue-integrable function, if: <math display="block"> \Pr [a \le X \le b] = \int_a^b f_X(x) \, dx .</math>

Hence, if <math>F_X</math> is the cumulative distribution function of <math>X</math>, then: <math display="block">F_X(x) = \int_{-\infty}^x f_X(u) \, du ,</math> and (if <math>f_X</math> is continuous at <math>x</math>) <math display="block"> f_X(x) = \frac{d}{dx} F_X(x) .</math>

Intuitively, one can think of <math>f_X(x) \, dx</math> as being the probability of <math>X</math> falling within the infinitesimal interval <math>[x,x+dx]</math>.

Formal definition

(This definition may be extended to any probability distribution using the measure-theoretic definition of probability.)

A random variable <math>X</math> with values in a measurable space <math>(\mathcal{X}, \mathcal{A})</math> (usually <math>\mathbb{R}^n</math> with the Borel sets as measurable subsets) has as probability distribution the pushforward measure X_∗P on <math>(\mathcal{X}, \mathcal{A})</math>: the density of <math>X</math> with respect to a reference measure <math>\mu</math> on <math>(\mathcal{X}, \mathcal{A})</math> is the Radon–Nikodym derivative: <math display="block">f = \frac{dX_*P}{d\mu} .</math>

That is, f is any measurable function with the property that: <math display="block">\Pr [X \in A ] = \int_{X^{-1} A} \, dP = \int_A f \, d\mu </math> for any measurable set <math>A \in \mathcal{A}.</math>

Discussion

In the continuous univariate case above, the reference measure is the Lebesgue measure. The probability mass function of a discrete random variable is the density with respect to the counting measure over the sample space (usually the set of integers, or some subset thereof).

It is not possible to define a density with reference to an arbitrary measure (e.g. one can not choose the counting measure as a reference for a continuous random variable). Furthermore, when it does exist, the density is almost unique, meaning that any two such densities coincide almost everywhere.

Further details

Unlike a probability, a probability density function can take on values greater than one; for example, the continuous uniform distribution on the interval Template:Closed-closed has probability density Template:Math for Template:Math and Template:Math elsewhere.

The standard normal distribution has probability density <math display="block">f(x) = \frac{1}{\sqrt{2\pi}}\, e^{-x^2/2}.</math>

If a random variable Template:Math is given and its distribution admits a probability density function Template:Math, then the expected value of Template:Math (if the expected value exists) can be calculated as <math display="block">\operatorname{E}[X] = \int_{-\infty}^\infty x\,f(x)\,dx.</math>

Not every probability distribution has a density function: the distributions of discrete random variables do not; nor does the Cantor distribution, even though it has no discrete component, i.e., does not assign positive probability to any individual point.

A distribution has a density function if its cumulative distribution function Template:Math is absolutely continuous. In this case: Template:Math is almost everywhere differentiable, and its derivative can be used as probability density: <math display="block">\frac{d}{dx}F(x) = f(x).</math>

If a probability distribution admits a density, then the probability of every one-point set Template:Math is zero; the same holds for finite and countable sets.

Two probability densities Template:Math and Template:Math represent the same probability distribution precisely if they differ only on a set of Lebesgue measure zero.

In the field of statistical physics, a non-formal reformulation of the relation above between the derivative of the cumulative distribution function and the probability density function is generally used as the definition of the probability density function. This alternate definition is the following:

If Template:Math is an infinitely small number, the probability that Template:Math is included within the interval Template:Open-open is equal to Template:Math, or: <math display="block">\Pr(t<X<t+dt) = f(t)\,dt.</math>

Link between discrete and continuous distributions

It is possible to represent certain discrete random variables as well as random variables involving both a continuous and a discrete part with a generalized probability density function using the Dirac delta function. (This is not possible with a probability density function in the sense defined above, it may be done with a distribution.) For example, consider a binary discrete random variable having the Rademacher distribution—that is, taking −1 or 1 for values, with probability Template:1/2 each. The density of probability associated with this variable is: <math display="block">f(t) = \frac{1}{2} (\delta(t+1)+\delta(t-1)).</math>

More generally, if a discrete variable can take Template:Mvar different values among real numbers, then the associated probability density function is: <math display="block">f(t) = \sum_{i=1}^n p_i\, \delta(t-x_i),</math> where <math>x_1, \ldots, x_n</math> are the discrete values accessible to the variable and <math>p_1, \ldots, p_n</math> are the probabilities associated with these values.

This substantially unifies the treatment of discrete and continuous probability distributions. The above expression allows for determining statistical characteristics of such a discrete variable (such as the mean, variance, and kurtosis), starting from the formulas given for a continuous distribution of the probability.

Families of densities

It is common for probability density functions (and probability mass functions) to be parametrized—that is, to be characterized by unspecified parameters. For example, the normal distribution is parametrized in terms of the mean and the variance, denoted by <math>\mu</math> and <math>\sigma^2</math> respectively, giving the family of densities <math display="block">

  f(x;\mu,\sigma^2) = \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2 }.

</math> Different values of the parameters describe different distributions of different random variables on the same sample space (the same set of all possible values of the variable); this sample space is the domain of the family of random variables that this family of distributions describes. A given set of parameters describes a single distribution within the family sharing the functional form of the density. From the perspective of a given distribution, the parameters are constants, and terms in a density function that contain only parameters, but not variables, are part of the normalization factor of a distribution (the multiplicative factor that ensures that the area under the density—the probability of something in the domain occurring— equals 1). This normalization factor is outside the kernel of the distribution.

Since the parameters are constants, reparametrizing a density in terms of different parameters to give a characterization of a different random variable in the family, means simply substituting the new parameter values into the formula in place of the old ones.

Densities associated with multiple variables

For continuous random variables Template:Math, it is also possible to define a probability density function associated to the set as a whole, often called joint probability density function. This density function is defined as a function of the Template:Mvar variables, such that, for any domain Template:Mvar in the Template:Mvar-dimensional space of the values of the variables Template:Math, the probability that a realisation of the set variables falls inside the domain Template:Mvar is <math display="block">\Pr \left( X_1,\ldots,X_n \isin D \right)

= \int_D f_{X_1,\ldots,X_n}(x_1,\ldots,x_n)\,dx_1 \cdots dx_n.</math>

If Template:Math is the cumulative distribution function of the vector Template:Math, then the joint probability density function can be computed as a partial derivative <math display="block"> f(x) = \left.\frac{\partial^n F}{\partial x_1 \cdots \partial x_n} \right|_x </math>

Marginal densities

For Template:Math, let Template:Math be the probability density function associated with variable Template:Math alone. This is called the marginal density function, and can be deduced from the probability density associated with the random variables Template:Math by integrating over all values of the other Template:Math variables: <math display="block">f_{X_i}(x_i) = \int f(x_1,\ldots,x_n)\, dx_1 \cdots dx_{i-1}\,dx_{i+1}\cdots dx_n .</math>

Independence

Continuous random variables Template:Math admitting a joint density are all independent from each other if <math display="block">f_{X_1,\ldots,X_n}(x_1,\ldots,x_n) = f_{X_1}(x_1)\cdots f_{X_n}(x_n).</math>

Corollary

If the joint probability density function of a vector of Template:Mvar random variables can be factored into a product of Template:Mvar functions of one variable <math display="block">f_{X_1,\ldots,X_n}(x_1,\ldots,x_n) = f_1(x_1)\cdots f_n(x_n),</math> (where each Template:Math is not necessarily a density) then the Template:Mvar variables in the set are all independent from each other, and the marginal probability density function of each of them is given by <math display="block">f_{X_i}(x_i) = \frac{f_i(x_i)}{\int f_i(x)\,dx}.</math>

Example

This elementary example illustrates the above definition of multidimensional probability density functions in the simple case of a function of a set of two variables. Let us call <math>\vec R</math> a 2-dimensional random vector of coordinates Template:Math: the probability to obtain <math>\vec R</math> in the quarter plane of positive Template:Math and Template:Math is <math display="block">\Pr \left( X > 0, Y > 0 \right)

= \int_0^\infty \int_0^\infty f_{X,Y}(x,y)\,dx\,dy.</math>

Function of random variables and change of variables in the probability density function

If the probability density function of a random variable (or vector) Template:Math is given as Template:Math, it is possible (but often not necessary; see below) to calculate the probability density function of some variable Template:Math. This is also called a "change of variable" and is in practice used to generate a random variable of arbitrary shape Template:Math using a known (for instance, uniform) random number generator.

It is tempting to think that in order to find the expected value Template:Math, one must first find the probability density Template:Math of the new random variable Template:Math. However, rather than computing <math display="block">\operatorname E\big(g(X)\big) = \int_{-\infty}^\infty y f_{g(X)}(y)\,dy, </math> one may find instead <math display="block">\operatorname E\big(g(X)\big) = \int_{-\infty}^\infty g(x) f_X(x)\,dx.</math>

The values of the two integrals are the same in all cases in which both Template:Math and Template:Math actually have probability density functions. It is not necessary that Template:Math be a one-to-one function. In some cases the latter integral is computed much more easily than the former. See Law of the unconscious statistician.

Scalar to scalar

Let <math> g: \Reals \to \Reals</math> be a monotonic function, then the resulting density function is<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block">f_Y(y) = f_X\big(g^{-1}(y)\big) \left| \frac{d}{dy} \big(g^{-1}(y)\big) \right|.</math>

Here Template:Math denotes the inverse function.

This follows from the fact that the probability contained in a differential area must be invariant under change of variables. That is, <math display="block">\left| f_Y(y)\, dy \right| = \left| f_X(x)\, dx \right|,</math> or <math display="block">f_Y(y) = \left| \frac{dx}{dy} \right| f_X(x) = \left| \frac{d}{dy} (x) \right| f_X(x) = \left| \frac{d}{dy} \big(g^{-1}(y)\big) \right| f_X\big(g^{-1}(y)\big) = {\left|\left(g^{-1}\right)'(y)\right|} \cdot f_X\big(g^{-1}(y)\big) .</math>

For functions that are not monotonic, the probability density function for Template:Mvar is <math display="block">\sum_{k=1}^{n(y)} \left| \frac{d}{dy} g^{-1}_{k}(y) \right| \cdot f_X\big(g^{-1}_{k}(y)\big),</math> where Template:Math is the number of solutions in Template:Mvar for the equation <math>g(x) = y</math>, and <math>g_k^{-1}(y)</math> are these solutions.

Vector to vector

Suppose Template:Math is an Template:Mvar-dimensional random variable with joint density Template:Math. If Template:Math, where Template:Math is a bijective, differentiable function, then Template:Math has density Template:Math: <math display="block"> p_{Y}(\mathbf{y}) = f\Bigl(G^{-1}(\mathbf{y})\Bigr) \left| \det\left[\left.\frac{dG^{-1}(\mathbf{z})}{d\mathbf{z}}\right|_{\mathbf{z}=\mathbf{y}}\right] \right|</math> with the differential regarded as the Jacobian of the inverse of Template:Math, evaluated at Template:Math.<ref>Template:Cite book</ref>

For example, in the 2-dimensional case Template:Math, suppose the transform Template:Math is given as Template:Math, Template:Math with inverses Template:Math, Template:Math. The joint distribution for y = (y₁, y₂) has density<ref>Template:Cite book</ref> <math display="block">p_{Y_1, Y_2}(y_1,y_2) = f_{X_1,X_2}\big(G_1^{-1}(y_1,y_2), G_2^{-1}(y_1,y_2)\big) \left\vert \frac{\partial G_1^{-1}}{\partial y_1} \frac{\partial G_2^{-1}}{\partial y_2} - \frac{\partial G_1^{-1}}{\partial y_2} \frac{\partial G_2^{-1}}{\partial y_1} \right\vert.</math>

Vector to scalar

Let <math> V: \R^n \to \R </math> be a differentiable function and <math> X </math> be a random vector taking values in <math> \R^n </math>, <math> f_X </math> be the probability density function of <math> X </math> and <math> \delta(\cdot) </math> be the Dirac delta function. It is possible to use the formulas above to determine <math> f_Y </math>, the probability density function of <math> Y = V(X) </math>, which will be given by <math display="block">f_Y(y) = \int_{\R^n} f_{X}(\mathbf{x}) \delta\big(y - V(\mathbf{x})\big) \,d \mathbf{x}.</math>

This result leads to the law of the unconscious statistician: <math display="block">\begin{align} \operatorname{E}_Y[Y] &=\int_{\R} y f_Y(y) \, dy \\ &= \int_{\R} y \int_{\R^n} f_X(\mathbf{x}) \delta\big(y - V(\mathbf{x})\big) \,d \mathbf{x} \,dy \\ &= \int_{{\mathbb R}^n} \int_{\mathbb R} y f_{X}(\mathbf{x}) \delta\big(y - V(\mathbf{x})\big) \, dy \, d \mathbf{x} \\ &= \int_{\mathbb R^n} V(\mathbf{x}) f_X(\mathbf{x}) \, d \mathbf{x}=\operatorname{E}_X[V(X)]. \end{align}</math>

Proof:

Let <math>Z</math> be a collapsed random variable with probability density function <math>p_Z(z) = \delta(z)</math> (i.e., a constant equal to zero). Let the random vector <math>\tilde{X}</math> and the transform <math>H</math> be defined as <math display="block">H(Z,X)=\begin{bmatrix} Z+V(X)\\ X\end{bmatrix}=\begin{bmatrix} Y\\ \tilde{X}\end{bmatrix}.</math>

It is clear that <math>H</math> is a bijective mapping, and the Jacobian of <math>H^{-1}</math> is given by: <math display="block">\frac{dH^{-1}(y,\tilde{\mathbf{x}})}{dy\,d\tilde{\mathbf{x}}}=\begin{bmatrix} 1 & -\frac{dV(\tilde{\mathbf{x}})}{d\tilde{\mathbf{x}}}\\ \mathbf{0}_{n\times1} & \mathbf{I}_{n\times n} \end{bmatrix},</math> which is an upper triangular matrix with ones on the main diagonal, therefore its determinant is 1. Applying the change of variable theorem from the previous section we obtain that <math display="block">f_{Y,X}(y,x) = f_X(\mathbf{x}) \delta\big(y - V(\mathbf{x})\big),</math> which if marginalized over <math>x</math> leads to the desired probability density function.

Sums of independent random variables

The probability density function of the sum of two independent random variables Template:Math and Template:Math, each of which has a probability density function, is the convolution of their separate density functions: <math display="block"> f_{U+V}(x) = \int_{-\infty}^\infty f_U(y) f_V(x - y)\,dy = \left( f_{U} * f_{V} \right) (x) </math>

It is possible to generalize the previous relation to a sum of N independent random variables, with densities Template:Math: <math display="block">f_{U_1 + \cdots + U}(x) = \left( f_{U_1} * \cdots * f_{U_N} \right) (x)</math>

This can be derived from a two-way change of variables involving Template:Math and Template:Math, similarly to the example below for the quotient of independent random variables.

Products and quotients of independent random variables

Given two independent random variables Template:Math and Template:Math, each of which has a probability density function, the density of the product Template:Math and quotient Template:Math can be computed by a change of variables.

Example: Quotient distribution

To compute the quotient Template:Math of two independent random variables Template:Math and Template:Math, define the following transformation: <math display="block">\begin{align} Y &= U/V \\[1ex] Z &= V \end{align}</math>

Then, the joint density Template:Math can be computed by a change of variables from U,V to Y,Z, and Template:Math can be derived by marginalizing out Template:Math from the joint density.

The inverse transformation is <math display="block">\begin{align} U &= YZ \\ V &= Z \end{align}</math>

The absolute value of the Jacobian matrix determinant <math>J(U,V\mid Y,Z)</math> of this transformation is: <math display="block"> \left| \det\begin{bmatrix} \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z} \\ \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} \end{bmatrix} \right| = \left| \det\begin{bmatrix} z & y \\ 0 & 1 \end{bmatrix} \right| = |z| . </math>

Thus: <math display="block">p(y,z) = p(u,v)\,J(u,v\mid y,z) = p(u)\,p(v)\,J(u,v\mid y,z) = p_U(yz)\,p_V(z)\, |z| .</math>

And the distribution of Template:Math can be computed by marginalizing out Template:Math: <math display="block">p(y) = \int_{-\infty}^\infty p_U(yz)\,p_V(z)\, |z| \, dz</math>

This method crucially requires that the transformation from U,V to Y,Z be bijective. The above transformation meets this because Template:Math can be mapped directly back to Template:Math, and for a given Template:Math the quotient Template:Math is monotonic. This is similarly the case for the sum Template:Math, difference Template:Math and product Template:Math.

Exactly the same method can be used to compute the distribution of other functions of multiple independent random variables.

Example: Quotient of two standard normals

Given two standard normal variables Template:Math and Template:Math, the quotient can be computed as follows. First, the variables have the following density functions: <math display="block">\begin{align} p(u) &= \frac{1}{\sqrt{2\pi}} e^{-{u^2}/{2}} \\[1ex] p(v) &= \frac{1}{\sqrt{2\pi}} e^{-{v^2}/{2}} \end{align}</math>

We transform as described above: <math display="block">\begin{align} Y &= U/V \\[1ex] Z &= V \end{align}</math>

This leads to: <math display="block">\begin{align} p(y) &= \int_{-\infty}^\infty p_U(yz)\,p_V(z)\, |z| \, dz \\[5pt] &= \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2} y^2 z^2} \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2} z^2} |z| \, dz \\[5pt] &= \int_{-\infty}^\infty \frac{1}{2\pi} e^{-\frac{1}{2}\left(y^2+1\right)z^2} |z| \, dz \\[5pt] &= 2\int_0^\infty \frac{1}{2\pi} e^{-\frac{1}{2}\left(y^2+1\right)z^2} z \, dz \\[5pt] &= \int_0^\infty \frac{1}{\pi} e^{-\left(y^2+1\right)u} \, du && u=\tfrac{1}{2}z^2\\[5pt] &= \left. -\frac{1}{\pi \left(y^2+1\right)} e^{-\left(y^2+1\right)u}\right|_{u=0}^\infty \\[5pt] &= \frac{1}{\pi \left(y^2+1\right)} \end{align}</math>

This is the density of a standard Cauchy distribution.

See also

• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Template:Annotated link
• Merging independent probability density functions
• Uses as position probability density:
  • Template:Annotated link
  • Template:Annotated link

Further reading

• Template:Cite book
• Template:Cite book
• Template:Cite book Chapters 7 to 9 are about continuous variables.

External links

• Template:Springer
• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:ProbabilityDensityFunction%7CProbabilityDensityFunction.html}} |title = Template:PAGENAMEBASE |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}