Cumulative distribution function

Summary

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 .[1]

Cumulative distribution function for the exponential distribution
Cumulative distribution function for the normal distribution

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) satisfying and .

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.

DefinitionEdit

The cumulative distribution function of a real-valued random variable   is the function given by[2]: p. 77 

 

 

 

 

 

(Eq.1)

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[2]: p. 84 

 

 

 

 

 

(Eq.2)

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   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 uses   and   instead of   and  , respectively.

The probability density function of a continuous random variable can be determined from the cumulative distribution function by differentiating[3] using the Fundamental Theorem of Calculus; i.e. given  ,

 
as long as the derivative exists.

The CDF of a continuous random variable   can be expressed as the integral of its probability density function   as follows:[2]: 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  .

PropertiesEdit

 
From top to bottom, the cumulative distribution function of a discrete probability distribution, continuous probability distribution, and a distribution which has both a continuous part and a discrete part.
 
Example of a cumulative distribution function with a countably infinite set of discontinuities.

Every cumulative distribution function   is non-decreasing[2]: p. 78  and right-continuous,[2]: p. 79  which makes it a càdlàg function. Furthermore,

 

Every function with these four 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   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  .

If   has finite L1-norm, that is, the expectation of   is finite, then the expectation is given by the Riemann–Stieltjes integral

 
and for any  ,
 
CDF plot with two red rectangles, illustrating   and  .

 
as shown in the diagram.

In particular, we have

 

ExamplesEdit

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

 

Suppose   is exponential distributed. Then the CDF of   is given by

 

Here λ > 0 is the parameter of the distribution, often called the rate parameter.

Suppose   is normal distributed. Then the CDF of   is given by

 

Here the parameter   is the mean or expectation of the distribution; and   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   is binomial distributed. Then the CDF of   is given by

 

Here   is the probability of success and the function denotes the discrete probability distribution of the number of successes in a sequence of   independent experiments, and   is the "floor" under  , i.e. the greatest integer less than or equal to  .

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 complementary cumulative distribution function (ccdf) or simply the tail distribution or exceedance, and is defined as

 

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   of the test statistic

 

In survival analysis,   is called the survival function and denoted  , 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[4]
     
  • 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.
  • For a random variable having an expectation,
     
    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
     

Folded cumulative distributionEdit

 
Example of the folded cumulative distribution for a normal distribution function with an expected value of 0 and a standard deviation of 1.

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,[5][6] that is

 

where   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[7]) and skewness 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  . This defines the inverse distribution function or quantile function.

Some distributions do not have a unique inverse (for example if   for all  , causing   to be constant). In this case, one may use the generalized inverse distribution function, which is defined as

 
  • 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
  2.  
  3.  
  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  .[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[citation needed].

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[2]: p. 89 

 

 

 

 

 

(Eq.3)

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  .

Example of joint cumulative distribution function:

For two continuous variables X and Y:

 

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:[8]

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 2 ≤ Y < 4 4 ≤ Y < 6 6 ≤ Y < 8 Y ≥ 8
X < 1 0 0 0 0 0
1 ≤ X < 3 0 0 0.1 0.1 0.2
3 ≤ X < 5 0 0 0.1 0.3 0.4
5 ≤ X < 7 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   random variables  , the joint CDF   is given by

 

 

 

 

 

(Eq.4)

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

 

PropertiesEdit

Every multivariate CDF is:

  1. Monotonically non-decreasing for each of its variables,
  2. Right-continuous in each of its variables,
  3.  
  4.  

Any function satisfying the above four properties is not a multivariate CDF, unlike in the single dimension case. For example, let   for   or   or   and let   otherwise. It is easy to see that the above conditions are met, and yet   is not a CDF since if it was, then   as explained below.

The probability that a point belongs to a hyperrectangle is analogous to the 1-dimensional case:[9]

 

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

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

  • Descriptive statistics
  • Distribution fitting
  • Ogive (statistics)
  • Modified half-normal distribution[10] with the pdf on   is given as  , where   denotes the Fox-Wright Psi function.

ReferencesEdit

  1. ^ Deisenroth, Marc Peter; Faisal, A. Aldo; Ong, Cheng Soon (2020). Mathematics for Machine Learning. Cambridge University Press. p. 181. ISBN 9781108455145.
  2. ^ 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.
  3. ^ Montgomery, Douglas C.; Runger, George C. (2003). Applied Statistics and Probability for Engineers (PDF). John Wiley & Sons, Inc. p. 104. ISBN 0-471-20454-4. Archived (PDF) from the original on 2012-07-30.
  4. ^ Zwillinger, Daniel; Kokoska, Stephen (2010). CRC Standard Probability and Statistics Tables and Formulae. CRC Press. p. 49. ISBN 978-1-58488-059-2.
  5. ^ Gentle, J.E. (2009). Computational Statistics. Springer. ISBN 978-0-387-98145-1. Retrieved 2010-08-06.[page needed]
  6. ^ Monti, K. L. (1995). "Folded Empirical Distribution Function Curves (Mountain Plots)". The American Statistician. 49 (4): 342–345. doi:10.2307/2684570. JSTOR 2684570.
  7. ^ 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.
  8. ^ "Joint Cumulative Distribution Function (CDF)". math.info. Retrieved 2019-12-11.
  9. ^ "Archived copy" (PDF). www.math.wustl.edu. Archived from the original (PDF) on 22 February 2016. Retrieved 13 January 2022.{{cite web}}: CS1 maint: archived copy as title (link)
  10. ^ Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021). "The Modified-Half-Normal distribution: Properties and an efficient sampling scheme". Communications in Statistics - Theory and Methods: 1–23. doi:10.1080/03610926.2021.1934700. ISSN 0361-0926. S2CID 237919587.

External linksEdit

  •   Media related to Cumulative distribution functions at Wikimedia Commons