11.3.5 Probability Mass Function from Cumulative Distribution Function

For a discrete random variable X , the cumulative distribution function F has jumps at each of the x i , and is constant between successive x s i ′ . The height of the jump at x i is f x i ( ) ; in this way the probability at x i can be retrieved from F . Fig. .

Suppose X is a discrete random variable taking the values x x x ,  such that x  and F x i ( ) is the distribution function. Then the probability mass function f x i ( ) is given by F x F x i i i , i = , , ,  Note The jump of a function F x ( ) at x is F a F a . Since F is non-decreasing and continuous to the right, the jump of a cumulative distribution function F is P X F x F x Here the jump (because of discontinuity) acts as a probability. That is, the set of discontinuities of a cumulative distribution function is at most countable!

Example . Find the probability mass function f x ( ) of the discrete random variable X whose cumulative distribution function F x ( ) is given by F x