Probability Mass Function - Formal Definition

Formal Definition

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

Note that f_X is defined for all real numbers, including those not in the image of X; indeed, f_X(x) = 0 for all x X(S). Essentially the same definition applies for a discrete multivariate random variable X: S → An, 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 f_X(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.