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If random variable X can take only a finite number k of different values x1,x2,…,xk or an infinite sequence of different values x1,x2,…, the random variable X has a discrete probability distribution. The probability mass function (pmf) of X is defined as the function f such that for every real number x,

(1.11)

If x is not one of the possible values of X, then f(x)=0. If the sequence x1,x2,… includes all the possible values of X, then ∑if(xi)=1. The cdf is given by

(1.12)