Expressing probability: cumulative distribution function and probability density function

Probability distribution of a random variable X is described with its cumulative distribution function (c.d.f.), which is the probability that X will be less than or equal to x:

The c.d.f. is also called the probability of nonexceedance of x. In flood frequency analysis, we are usually interested in the probability of exceedance of x:

For discrete random variables it makes sense to define probabilities that X will be equal to an integer value x, P{X = x}. For continuous random variables this is not possible since they can take any real value (e.g. the magnitude of floods is described with positive real values, X ³ 0). Therefore the continuous random variables are described with the probability density function (p.d.f.), which is the derivative of the cumulative distribution function:

The cumulative distribution function can also be expressed as the integral of the probability density function:

where u is a dummy variable for integration. Graphically (see figure on the right), probability of nonexceedance represents:

shaded area below p.d.f., and

ordinate of c.d.f.

Similarly, the probability of exceedance is:

Probability that X will have values in some interval (x_{i}_{–1}, x_{i}) can also be expressed as the integral of the p.d.f.:

and is also equal to difference of c.d.f. evaluated at x_{i}_{–1} and x_{i}.