Probability Mass Function - Formal Definition

Formal Definition

Suppose that X: SA (A R) is a discrete random variable defined on a sample space S. Then the probability mass function fX: A → for X is defined as

Note that fX is defined for all real numbers, including those not in the image of X; indeed, fX(x) = 0 for all x X(S). Essentially the same definition applies for a discrete multivariate random variable X: SAn, with scalar values being replaced by vector values.

The total probability for all X must equal 1

Since the image of X is countable, the probability mass function fX(x) is zero for all but a countable number of values of x. The discontinuity of probability mass functions is related to the fact that the cumulative distribution function of a discrete random variable is also discontinuous. Where it is differentiable, the derivative is zero, just as the probability mass function is zero at all such points.

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