Cumulative Distribution Function - Definition

Definition

For every real number x, the cumulative distribution function of a real-valued random variable 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], where a < b, is therefore

Here the notation (a, b], indicates a semi-closed interval.

If treating several random variables X, Y, ... etc. the corresponding letters are used as subscripts while, if treating only one, the subscript is omitted. 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. 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 X can be defined in terms of its probability density function ƒ as follows:

Note that 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 is important for discrete distributions. 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.

In the case of a random variable X which has distribution having a discrete component at a value x₀,

where F(x₀-) denotes the limit from the left of F at x₀: i.e. lim F(y) as y increases towards x₀.