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.

Read more about this topic:  Probability Mass Function

Famous quotes containing the words formal and/or definition:

    The bed is now as public as the dinner table and governed by the same rules of formal confrontation.
    Angela Carter (1940–1992)

    Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.
    The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)