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 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: 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 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.
Read more about this topic: Probability Mass Function
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