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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)

so P(x) (when it exists) is simply the derivative of the distribution function

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

Similarly, the distribution function is related to a discrete probability P(x) by

D(x)=P(X<=x)
(4)
=sum_(X<=x)P(x).
(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)

The probability content of a closed region can be found much more efficiently than by direct integration of the probability density function P(x) by appropriate evaluation of the distribution function at all possible extrema defined on the region (Rose and Smith 1996; 2002, p. 193). For example, for a bivariate distribution function D(x,y), the probability content 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)

but can be computed much more efficiently 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. The method of finding the value at which the distribution is a maximum is known as the maximum likelihood method.


See also

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

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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.

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Distribution Function

Cite this as:

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

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