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GeezerVison is showing mathworld.wolfram.com — Distribution Function -- from Wolfram MathWorld

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Distribution Function

The distribution function D(x), also called the cumulative distribution function (CDF) or cumulative frequency function, describes the probability that a variate X takes on a value less than or equal to a number x. The distribution function is sometimes also denoted F(x) (Evans et al. 2000, p. 6).

The distribution function is therefore related to a continuous probability density function P(x) by

D(x)=P(X<=x)
(1)
=int_(-infty)^xP(xi)dxi,
(2)

where integration may be restricted to the support of the probability density function, since P(xi)=0 outside that support. Thus P(x) (when it exists) is the derivative of the distribution function

P(x)=D^'(x).
(3)

For a discrete distribution with probability mass function p(t)=P(X=t), the distribution function is

D(x)=P(X<=x)
(4)
=sum_(t<=x)p(t).
(5)

There exist distributions that are neither continuous nor discrete.

A joint distribution function can be defined if outcomes are dependent on two parameters:

D(x,y)=P(X<=x,Y<=y)
(6)
D_x(x)=D(x,infty)
(7)
D_y(y)=D(infty,y).
(8)

Similarly, a multivariate distribution function can be defined if outcomes depend on n parameters:

D(a_1,...,a_n)=P(x_1<=a_1,...,x_n<=a_n).
(9)

For a bivariate distribution with joint probability density function P(x,y) and distribution function D(x,y), the probability of a rectangle can be evaluated using the distribution function (Rose and Smith 1996; Rose and Smith 2002, p. 193). The probability in the region x_1<=x<=x_2, y_1<=y<=y_2 is given by

P(x_1<=x<=x_2,y_1<=y<=y_2)=int_(x_1)^(x_2)int_(y_1)^(y_2)P(x,y)dydx,
(10)

and can also be computed using

P(x_1<=x<=x_2,y_1<=y<=y_2)=D(x_1,y_1)-D(x_1,y_2)-D(x_2,y_1)+D(x_2,y_2).
(11)

Given a continuous P(x), assume you wish to generate numbers distributed as P(x) using a random number generator. If the random number generator yields a uniformly distributed value y_i in [0,1] for each trial i, then compute

D(x)=int^xP(x^')dx^'.
(12)

The formula connecting y_i with a variable distributed as P(x) is then

x_i=D^(-1)(y_i),
(13)

where D^(-1)(x) is the inverse function of D(x). For example, if P(x) were a normal distribution so that

D(x)=1/2[1+erf((x-mu)/(sigmasqrt(2)))],
(14)

then

x_i=sigmasqrt(2)erf^(-1)(2y_i-1)+mu.
(15)

A distribution with constant variance of y for all values of x is known as a homoscedastic distribution. A value at which the probability density function attains its maximum is a mode.

Cumulative distribution functions are computed in the Wolfram Language using CDF.

See also

Cumulative Count, Cumulative Frequency, Probability Density Function, Stochastic Dominance, Survival Function, Variate

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  • cumulative distribution function of the standard normal distribution
  • cumulative distribution function gaussian mean=mu standard deviation=sigma
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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Probability Functions." Ch. 26 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 925-964, 1972.Evans, M.; Hastings, N.; and Peacock, B. Statistical Distributions, 3rd ed. New York: Wiley, pp. 6-8, 2000.Iyanaga, S. and Kawada, Y. (Eds.). "Distribution of Typical Random Variables." Appendix A, Table 22 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1483-1486, 1980.Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 92-94, 1984.Rose, C. and Smith, M. D. "The Multivariate Normal Distribution." Mathematica J. 6, 32-37, 1996.Rose, C. and Smith, M. D. Mathematical Statistics with Mathematica. New York: Springer-Verlag, 2002.

Referenced on Wolfram|Alpha

Distribution Function

Cite this as:

Weisstein, Eric W. "Distribution Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DistributionFunction.html

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