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We say that a random variable X has a continuous distribution or that X is a continuous random variable if there exists a nonnegative function f, defined on the real line, such that for every interval of real numbers (bounded or unbounded), the probability that X takes a value in an interval [a, b] is the integral of f over that interval, that is,

(1.19)

If X has a continuous distribution, the function f will be the probability density function (pdf) of X. The pdf must satisfy the following requirements:

(1.20)

The cdf of a continuous distribution is given by

(1.21)

The mean, μ, and variance, σ2, of the continuous random variable are calculated by

(1.22)