Cumulative Distribution Function

In probability theory, the cumulative distribution function (cdf), also called probability distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X. For every real number x, the CDF of X is given by

where the right-hand side represents the probability that the random variable X takes on a value less than or equal to x. The probability that X lies in the interval [a, b] is therefore F(b) - F(a) if a < b. It is conventional to use a capital F for a cumulative distribution function, in contrast to the lower-case f used for probability density functions and probability mass functions.

The CDF of X can be defined in terms of the probability density function f as follows:

Ironically, the three word phrase "cumulative distribution function" is regarded as an anachronism by professional probabilisits and many academic statisticians. To them, the adjective "cumulative" is redundant.

Note that in the definition above, the "less or equal" sign, '=' is a convention, but it is an important and universally used one. The proper use of tables of the Binomial and Poisson distributions depend upon this convention. Moreover, important formulas like Levy's inversion formula for the characteristic function also rely on the "less or equal" formulation.

The complete, up-to-date and editable article about Cumulative Distribution Function can be found at Wikipedia: Cumulative Distribution Function
http://en.wikipedia.org/wiki/Cumulative_Distribution_Function