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Cumulative distribution function for the Exponential distribution
Cumulative distribution function for the Normal distribution

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable , or just distribution function of , evaluated at , is the probability that will take a value less than or equal to .

In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to . Cumulative distribution functions are also used to specify the distribution of multivariate random variables.



The cumulative distribution function of a real-valued random variable   is the function given by[1]:p. 77







where the right-hand side represents the probability that the random variable   takes on a value less than or equal to  . The probability that   lies in the semi-closed interval  , where  , is therefore[1]:p. 84








If treating several random variables   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   for a cumulative distribution function, in contrast to the lower-case   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.

The CDF of a continuous random variable   can be expressed as the integral of its probability density function   as follows:[1]:p. 86


In the case of a random variable   which has distribution having a discrete component at a value  ,


If   is continuous at  , this equals zero and there is no discrete component at  .



Every cumulative distribution function   is non-decreasing[1]:p. 78 and right-continuous[1]:p. 79, which makes it a càdlàg function. Furthermore,



If   is a purely discrete random variable, then it attains values   with probability  , and the CDF of   will be discontinuous at the points  :


If the CDF   of a real valued random variable   is continuous, then   is a continuous random variable; if furthermore   is absolutely continuous, then there exists a Lebesgue-integrable function   such that


for all real numbers   and  . The function   is equal to the derivative of   almost everywhere, and it is called the probability density function of the distribution of  .


As an example, suppose   is uniformly distributed on the unit interval  . Then the CDF of   is given by


Suppose instead that   takes only the discrete values 0 and 1, with equal probability. Then the CDF of   is given by


Derived functionsEdit

Complementary cumulative distribution function (tail distribution)Edit





In survival analysis,   is called the survival function and denoted  , while the term reliability function is common in engineering.

  • As  , and in fact   provided that   is finite.
Proof:[citation needed] Assuming   has a density function  , for any  
Then, on recognizing   and rearranging terms,
as claimed.

Folded cumulative distributionEdit


While the plot of a cumulative distribution 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,[3][4] thus using two scales, one for the upslope and another for the downslope. This form of illustration emphasises the median and dispersion (specifically, the mean absolute deviation from the median[5]) of the distribution or of the empirical results.

Inverse distribution function (quantile function)Edit

If the CDF F is strictly increasing and continuous then   is the unique real number   such that  . In such a case, this defines the inverse distribution function or quantile function.

Some distributions do not have a unique inverse (for example in the case where   for all  , causing   to be constant). This problem can be solved by defining, for  , the generalized inverse distribution function:

  • Example 1: The median is  .
  • Example 2: Put  . Then we call   the 95th percentile.

Some useful properties of the inverse cdf (which are also preserved in the definition of the generalized inverse distribution function) are:

  1.   is nondecreasing
  4.   if and only if  
  5. If   has a   distribution then   is distributed as  . This is used in random number generation using the inverse transform sampling-method.
  6. If   is a collection of independent  -distributed random variables defined on the same sample space, then there exist random variables   such that   is distributed as   and   with probability 1 for all  .


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  , the joint CDF   is given by[1]:p. 89







where the right-hand side represents the probability that the random variable   takes on a value less than or equal to   and that   takes on a value less than or equal to  .

Definition for more than two random variablesEdit

For   random variables  , the joint CDF   is given by







Interpreting the   random variables as a random vector   yields a shorter notation:



Every multivariate CDF is:

  1. Monotonically non-decreasing for each of its variables,
  2. Right-continuous in each of its variables,

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   make no sense. However expressions of the form   make sense. Therefore, we define the cumulative distribution of a complex random variables via the joint distribution of their real and imaginary parts:


Complex random vectorEdit

Generalization of Eq.4 yields


as definition for the CDS of a complex random vector  .

Use in statistical analysisEdit


Kolmogorov–Smirnov and Kuiper's testsEdit


See alsoEdit


  1. ^ a b c d e f Park, Kun Il (2018). Fundamentals of Probability and Stochastic Processes with Applications to Communications. Springer. ISBN 978-3-319-68074-3.
  4. ^ Monti, K.L. (1995). "Folded Empirical Distribution Function Curves (Mountain Plots)". The American Statistician. 49 (4): 342–345. doi:10.2307/2684570. JSTOR 2684570.
  5. ^ Xue, J. H.; Titterington, D. M. (2011). "The p-folded cumulative distribution function and the mean absolute deviation from the p-quantile" (PDF). Statistics & Probability Letters. 81 (8): 1179–1182. doi:10.1016/j.spl.2011.03.014.<

External linksEdit