Cumulative distribution function
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In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable <math>X</math>, or just distribution function of <math>X</math>, evaluated at <math>x</math>, is the probability that <math>X</math> will take a value less than or equal to <math>x</math>.<ref>Template:Cite book</ref>
Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by a right-continuous monotone increasing function (a càdlàg function) <math>F \colon \mathbb R \rightarrow [0,1]</math> satisfying <math>\lim_{x\rightarrow-\infty}F(x)=0</math> and <math>\lim_{x\rightarrow\infty}F(x)=1</math>.
In the case of a scalar continuous distribution, it gives the area under the probability density function from negative infinity to <math>x</math>. Cumulative distribution functions are also used to specify the distribution of multivariate random variables.
DefinitionEdit
The cumulative distribution function of a real-valued random variable <math>X</math> is the function given by<ref name=KunIlPark>Template:Cite book</ref>Template:Rp Template:Equation box 1 where the right-hand side represents the probability that the random variable <math>X</math> takes on a value less than or equal to <math>x</math>.
The probability that <math>X</math> lies in the semi-closed interval <math>(a,b]</math>, where <math>a < b</math>, is therefore<ref name=KunIlPark/>Template:Rp
In the definition above, the "less than or equal to" sign, "≤", is a convention, not a universally used one (e.g. Hungarian literature uses "<"), but the distinction is important for discrete distributions. The proper use of tables of the binomial and Poisson distributions depends upon this convention. Moreover, important formulas like Paul Lévy's inversion formula for the characteristic function also rely on the "less than or equal" formulation.
If treating several random variables <math>X, Y, \ldots</math> etc. the corresponding letters are used as subscripts while, if treating only one, the subscript is usually omitted. It is conventional to use a capital <math>F</math> for a cumulative distribution function, in contrast to the lower-case <math>f</math> used for probability density functions and probability mass functions. This applies when discussing general distributions: some specific distributions have their own conventional notation, for example the normal distribution uses <math>\Phi</math> and <math>\phi</math> instead of <math>F</math> and <math>f</math>, respectively.
The probability density function of a continuous random variable can be determined from the cumulative distribution function by differentiating<ref>Template:Cite book</ref> using the Fundamental Theorem of Calculus; i.e. given <math>F(x)</math>, <math display="block">f(x) = \frac{dF(x)}{dx}</math> as long as the derivative exists.
The CDF of a continuous random variable <math>X</math> can be expressed as the integral of its probability density function <math>f_X</math> as follows:<ref name="KunIlPark" />Template:Rp <math display="block">F_X(x) = \int_{-\infty}^x f_X(t) \, dt.</math>
In the case of a random variable <math>X</math> which has distribution having a discrete component at a value <math>b</math>, <math display="block">\operatorname{P}(X=b) = F_X(b) - \lim_{x \to b^-} F_X(x).</math>
If <math>F_X</math> is continuous at <math>b</math>, this equals zero and there is no discrete component at <math>b</math>.
PropertiesEdit
Every cumulative distribution function <math>F_X</math> is non-decreasing<ref name=KunIlPark/>Template:Rp and right-continuous,<ref name=KunIlPark/>Template:Rp which makes it a càdlàg function. Furthermore, <math display="block">\lim_{x \to -\infty} F_X(x) = 0, \quad \lim_{x \to +\infty} F_X(x) = 1.</math>
Every function with these three properties is a CDF, i.e., for every such function, a random variable can be defined such that the function is the cumulative distribution function of that random variable.
If <math>X</math> is a purely discrete random variable, then it attains values <math>x_1,x_2,\ldots</math> with probability <math>p_i = p(x_i)</math>, and the CDF of <math>X</math> will be discontinuous at the points <math>x_i</math>: <math display="block">F_X(x) = \operatorname{P}(X\leq x) = \sum_{x_i \leq x} \operatorname{P}(X = x_i) = \sum_{x_i \leq x} p(x_i).</math>
If the CDF <math>F_X</math> of a real valued random variable <math>X</math> is continuous, then <math>X</math> is a continuous random variable; if furthermore <math>F_X</math> is absolutely continuous, then there exists a Lebesgue-integrable function <math>f_X(x)</math> such that <math display="block">F_X(b)-F_X(a) = \operatorname{P}(a< X\leq b) = \int_a^b f_X(x)\,dx</math> for all real numbers <math>a</math> and <math>b</math>. The function <math>f_X</math> is equal to the derivative of <math>F_X</math> almost everywhere, and it is called the probability density function of the distribution of <math>X</math>.
If <math>X</math> has finite L1-norm, that is, the expectation of <math>|X|</math> is finite, then the expectation is given by the Riemann–Stieltjes integral <math display="block"> \mathbb E[X] = \int_{-\infty}^\infty t\,dF_X(t) </math>
F(x)_and_x(1-F(x)).png){kind=link}
and for any <math>x \geq 0</math>, <math display="block"> x (1-F_X(x)) \leq \int_x^{\infty} t\,dF_X(t) </math> as well as <math display="block"> x F_X(-x) \leq \int_{-\infty}^{-x} (-t)\,dF_X(t) </math> as shown in the diagram (consider the areas of the two red rectangles and their extensions to the right or left up to the graph of <math>F_X</math>).Template:Clarify In particular, we have <math display="block"> \lim_{x \to -\infty} x F_X(x) = 0, \quad \lim_{x \to +\infty} x (1-F_X(x)) = 0. </math> In addition, the (finite) expected value of the real-valued random variable <math>X</math> can be defined on the graph of its cumulative distribution function as illustrated by the drawing in the definition of expected value for arbitrary real-valued random variables.
ExamplesEdit
As an example, suppose <math>X</math> is uniformly distributed on the unit interval <math>[0,1]</math>.
Then the CDF of <math>X</math> is given by <math display="block">F_X(x) = \begin{cases} 0 &:\ x < 0\\ x &:\ 0 \le x \le 1\\ 1 &:\ x > 1 \end{cases}</math>
Suppose instead that <math>X</math> takes only the discrete values 0 and 1, with equal probability.
Then the CDF of <math>X</math> is given by <math display="block">F_X(x) = \begin{cases} 0 &:\ x < 0\\ 1/2 &:\ 0 \le x < 1\\ 1 &:\ x \ge 1 \end{cases}</math>
Suppose <math>X</math> is exponential distributed. Then the CDF of <math>X</math> is given by <math display="block">F_X(x;\lambda) = \begin{cases} 1-e^{-\lambda x} & x \ge 0, \\ 0 & x < 0. \end{cases}</math>
Here λ > 0 is the parameter of the distribution, often called the rate parameter.
Suppose <math>X</math> is normal distributed. Then the CDF of <math>X</math> is given by <math display="block">F(t;\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}} \int_{-\infty}^t \exp \left( -\frac{(x - \mu)^2}{2\sigma^2} \right)\, dx. </math>
Here the parameter <math>\mu</math> is the mean or expectation of the distribution; and <math>\sigma</math> is its standard deviation.
A table of the CDF of the standard normal distribution is often used in statistical applications, where it is named the standard normal table, the unit normal table, or the Z table.
Suppose <math>X</math> is binomial distributed. Then the CDF of <math>X</math> is given by <math display="block">F(k;n,p) = \Pr(X\leq k) = \sum _{i=0}^{\lfloor k\rfloor }{n \choose i} p^{i} (1-p)^{n-i}</math>
Here <math>p</math> is the probability of success and the function denotes the discrete probability distribution of the number of successes in a sequence of <math>n</math> independent experiments, and <math>\lfloor k\rfloor</math> is the "floor" under <math>k</math>, i.e. the greatest integer less than or equal to <math>k</math>.
Derived functionsEdit
Complementary cumulative distribution function (tail distribution)Edit
Sometimes, it is useful to study the opposite question and ask how often the random variable is above a particular level. This is called the Template:Visible anchor (Template:Visible anchor) or simply the Template:Visible anchor or Template:Visible anchor, and is defined as <math display="block">\bar F_X(x) = \operatorname{P}(X > x) = 1 - F_X(x).</math>
This has applications in statistical hypothesis testing, for example, because the one-sided p-value is the probability of observing a test statistic at least as extreme as the one observed. Thus, provided that the test statistic, T, has a continuous distribution, the one-sided p-value is simply given by the ccdf: for an observed value <math>t</math> of the test statistic <math display="block">p= \operatorname{P}(T \ge t) = \operatorname{P}(T > t) = 1 - F_T(t).</math>
In survival analysis, <math>\bar F_X(x)</math> is called the survival function and denoted <math>S(x)</math>, while the term reliability function is common in engineering.
- Properties
- For a non-negative continuous random variable having an expectation, Markov's inequality states that<ref name="ZK">Template:Cite book</ref> <math display="block">\bar F_X(x) \leq \frac{\operatorname{E}(X)}{x} .</math>
- As <math>x \to \infty, \bar F_X(x) \to 0</math>, and in fact <math>\bar F_X(x) = o(1/x)</math> provided that <math>\operatorname{E}(X)</math> is finite.
Proof:Template:Citation needed
Assuming <math>X</math> has a density function <math>f_X</math>, for any <math>c > 0</math> <math display="block">
\operatorname{E}(X) = \int_0^\infty x f_X(x) \, dx \geq \int_0^c x f_X(x) \, dx + c\int_c^\infty f_X(x) \, dx </math> Then, on recognizing <math display="block">\bar F_X(c) = \int_c^\infty f_X(x) \, dx</math> and rearranging terms, <math display="block"> 0 \leq c\bar F_X(c) \leq \operatorname{E}(X) - \int_0^c x f_X(x) \, dx \to 0 \text{ as } c \to \infty </math> as claimed.
- For a random variable having an expectation, <math display="block">\operatorname{E}(X) = \int_0^\infty \bar F_X(x) \, dx - \int_{-\infty}^0 F_X(x) \, dx</math> and for a non-negative random variable the second term is 0.
If the random variable can only take non-negative integer values, this is equivalent to <math display="block">\operatorname{E}(X) = \sum_{n=0}^\infty \bar F_X(n).</math>
Folded cumulative distributionEdit
While the plot of a cumulative distribution <math>F</math> often has an S-like shape, an alternative illustration is the folded cumulative distribution or mountain plot, which folds the top half of the graph over,<ref name="Gentle">Template:Cite bookTemplate:Page needed</ref><ref name="Monti"> Template:Cite journal</ref> that is
- <math>F_\text{fold}(x)=F(x)1_{\{F(x)\leq 0.5\}}+(1-F(x))1_{\{F(x)>0.5\}}</math>
where <math>1_{\{A\}}</math> denotes the indicator function and the second summand is the survivor function, thus using two scales, one for the upslope and another for the downslope. This form of illustration emphasises the median, dispersion (specifically, the mean absolute deviation from the median<ref>Template:Cite journal</ref>) and skewness of the distribution or of the empirical results.
Inverse distribution function (quantile function)Edit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} If the CDF F is strictly increasing and continuous then <math> F^{-1}( p ), p \in [0,1], </math> is the unique real number <math> x </math> such that <math> F(x) = p </math>. This defines the inverse distribution function or quantile function.
Some distributions do not have a unique inverse (for example if <math>f_X(x)=0</math> for all <math>a<x<b</math>, causing <math>F_X</math> to be constant). In this case, one may use the generalized inverse distribution function, which is defined as
- <math>
F^{-1}(p) = \inf \{x \in \mathbb{R}: F(x) \geq p \}, \quad \forall p \in [0,1]. </math>
- Example 1: The median is <math>F^{-1}( 0.5 )</math>.
- Example 2: Put <math> \tau = F^{-1}( 0.95 ) </math>. Then we call <math> \tau </math> the 95th percentile.
Some useful properties of the inverse cdf (which are also preserved in the definition of the generalized inverse distribution function) are:
- <math>F^{-1}</math> is nondecreasing<ref>Template:Cite book</ref>
- <math>F^{-1}(F(x)) \leq x</math>
- <math>F(F^{-1}(p)) \geq p</math>
- <math>F^{-1}(p) \leq x</math> if and only if <math>p \leq F(x)</math>
- If <math>Y</math> has a <math>U[0, 1]</math> distribution then <math>F^{-1}(Y)</math> is distributed as <math>F</math>. This is used in random number generation using the inverse transform sampling-method.
- If <math>\{X_\alpha\}</math> is a collection of independent <math>F</math>-distributed random variables defined on the same sample space, then there exist random variables <math>Y_\alpha</math> such that <math>Y_\alpha</math> is distributed as <math>U[0,1]</math> and <math>F^{-1}(Y_\alpha) = X_\alpha</math> with probability 1 for all <math>\alpha</math>.Template:Citation needed
The inverse of the cdf can be used to translate results obtained for the uniform distribution to other distributions.
Empirical distribution functionEdit
The empirical distribution function is an estimate of the cumulative distribution function that generated the points in the sample. It converges with probability 1 to that underlying distribution. A number of results exist to quantify the rate of convergence of the empirical distribution function to the underlying cumulative distribution function.<ref>Template:Cite journal</ref>
Multivariate caseEdit
Definition for two random variablesEdit
When dealing simultaneously with more than one random variable the joint cumulative distribution function can also be defined. For example, for a pair of random variables <math>X,Y</math>, the joint CDF <math>F_{XY}</math> is given by<ref name=KunIlPark/>Template:Rp
Template:Equation box 1 where the right-hand side represents the probability that the random variable <math>X</math> takes on a value less than or equal to <math>x</math> and that <math>Y</math> takes on a value less than or equal to <math>y</math>.
Example of joint cumulative distribution function:
For two continuous variables X and Y: <math display="block"> \Pr(a < X < b \text{ and } c < Y < d) = \int_a^b \int_c^d f(x,y) \, dy \, dx;</math>
For two discrete random variables, it is beneficial to generate a table of probabilities and address the cumulative probability for each potential range of X and Y, and here is the example:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
given the joint probability mass function in tabular form, determine the joint cumulative distribution function.
| Y = 2 | Y = 4 | Y = 6 | Y = 8 | |
| X = 1 | 0 | 0.1 | 0 | 0.1 |
| X = 3 | 0 | 0 | 0.2 | 0 |
| X = 5 | 0.3 | 0 | 0 | 0.15 |
| X = 7 | 0 | 0 | 0.15 | 0 |
Solution: using the given table of probabilities for each potential range of X and Y, the joint cumulative distribution function may be constructed in tabular form:
| Y < 2 | Y ≤ 2 | Y ≤ 4 | Y ≤ 6 | Y ≤ 8 | |
| X < 1 | 0 | 0 | 0 | 0 | 0 |
| X ≤ 1 | 0 | 0 | 0.1 | 0.1 | 0.2 |
| X ≤ 3 | 0 | 0 | 0.1 | 0.3 | 0.4 |
| X ≤ 5 | 0 | 0.3 | 0.4 | 0.6 | 0.85 |
| X ≤ 7 | 0 | 0.3 | 0.4 | 0.75 | 1 |
Definition for more than two random variablesEdit
For <math>N</math> random variables <math>X_1,\ldots,X_N</math>, the joint CDF <math>F_{X_1,\ldots,X_N}</math> is given by
Interpreting the <math>N</math> random variables as a random vector <math>\mathbf{X} = (X_1, \ldots, X_N)^T</math> yields a shorter notation: <math display="block">F_{\mathbf{X}}(\mathbf{x}) = \operatorname{P}(X_1 \leq x_1,\ldots,X_N \leq x_N)</math>
PropertiesEdit
Every multivariate CDF is:
- Monotonically non-decreasing for each of its variables,
- Right-continuous in each of its variables,
- <math>0\leq F_{X_1 \ldots X_n}(x_1,\ldots,x_n)\leq 1,</math>
- <math>\lim_{x_1,\ldots,x_n \to+\infty}F_{X_1 \ldots X_n}(x_1,\ldots,x_n)=1 </math> and <math> \lim_{x_i\to-\infty}F_{X_1 \ldots X_n}(x_1,\ldots,x_n)=0,</math> for all Template:Mvar.
Not every function satisfying the above four properties is a multivariate CDF, unlike in the single dimension case. For example, let <math>F(x,y)=0</math> for <math>x<0</math> or <math>x+y<1</math> or <math>y<0</math> and let <math>F(x,y)=1</math> otherwise. It is easy to see that the above conditions are met, and yet <math>F</math> is not a CDF since if it was, then <math display="inline">\operatorname{P}\left(\frac{1}{3} < X \leq 1, \frac{1}{3} < Y \leq 1\right)=-1</math> as explained below.
The probability that a point belongs to a hyperrectangle is analogous to the 1-dimensional case:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block">F_{X_1,X_2}(a, c) + F_{X_1,X_2}(b, d) - F_{X_1,X_2}(a, d) - F_{X_1,X_2}(b, c) = \operatorname{P}(a < X_1 \leq b, c < X_2 \leq d) = \int \cdots</math>
Complex caseEdit
Complex random variableEdit
The generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form <math> P(Z \leq 1+2i) </math> make no sense. However expressions of the form <math> P(\Re{(Z)} \leq 1, \Im{(Z)} \leq 3) </math> make sense. Therefore, we define the cumulative distribution of a complex random variables via the joint distribution of their real and imaginary parts: <math display="block"> F_Z(z) = F_{\Re{(Z)},\Im{(Z)}}(\Re{(z)},\Im{(z)}) = P(\Re{(Z)} \leq \Re{(z)} , \Im{(Z)} \leq \Im{(z)}). </math>
Complex random vectorEdit
Generalization of Template:EquationNote yields <math display="block">F_{\mathbf{Z}}(\mathbf{z}) = F_{\Re{(Z_1)},\Im{(Z_1)}, \ldots, \Re{(Z_n)},\Im{(Z_n)}}(\Re{(z_1)}, \Im{(z_1)},\ldots,\Re{(z_n)}, \Im{(z_n)}) = \operatorname{P}(\Re{(Z_1)} \leq \Re{(z_1)},\Im{(Z_1)} \leq \Im{(z_1)},\ldots,\Re{(Z_n)} \leq \Re{(z_n)},\Im{(Z_n)} \leq \Im{(z_n)})</math> as definition for the CDS of a complex random vector <math>\mathbf{Z} = (Z_1,\ldots,Z_N)^T</math>.
Use in statistical analysisEdit
The concept of the cumulative distribution function makes an explicit appearance in statistical analysis in two (similar) ways. Cumulative frequency analysis is the analysis of the frequency of occurrence of values of a phenomenon less than a reference value. The empirical distribution function is a formal direct estimate of the cumulative distribution function for which simple statistical properties can be derived and which can form the basis of various statistical hypothesis tests. Such tests can assess whether there is evidence against a sample of data having arisen from a given distribution, or evidence against two samples of data having arisen from the same (unknown) population distribution.
Kolmogorov–Smirnov and Kuiper's testsEdit
The Kolmogorov–Smirnov test is based on cumulative distribution functions and can be used to test to see whether two empirical distributions are different or whether an empirical distribution is different from an ideal distribution. The closely related Kuiper's test is useful if the domain of the distribution is cyclic as in day of the week. For instance Kuiper's test might be used to see if the number of tornadoes varies during the year or if sales of a product vary by day of the week or day of the month.
See alsoEdit
ReferencesEdit
External linksEdit
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