Cumulative Probability Function

The cumulative probability function of a random variable (discrete or continuous) is a function whose domain is similar to that of the probability mass or density function, but whose range is the set of probabilities associated with the possibility that the random variable will assume a value that is less than or equal to the values in the domain. The cumulative probability function of a random variable X is denoted by F_X(x) and is defined as

F_X(x) = P (X £ x)

For a discrete random variable X with a probability mass function of p_X(x), its cumulative probability function F_X(x) is given by

NOTE: A continuous random variable Y with a probability density function of f_y(y) has a cumulative distribution function F_y(y) given by

EX. The discrete random variable X that assumes the value of the roll of two dice has the following cumulative probability distribution.

Note that for x = 2, 3, 4, ..... , 12