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The (cumulative) distribution function, or probability distribution function, F(x) is the mathematical equation that describes the probability that a variable X is less that or equal to x, i.e.

     F(x) = P(Xx)                    for all x

where P(Xx) means the probability of the event Xx.

A cumulative distribution function has the following properties:

1.    F(x) is always non-decreasing, i.e.  \frac{d}{dx} F(x)\geq 0

2.    F(x) = 0 at x = -∞ or minimum

     F(x) = 1 at x = ∞ or maximum

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