. . 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 Also find (i) P X < and (ii) P X .
Since X is a discrete random variable, from the given data, X takes on the values , and . For discrete random variable X, by definition, we have f x ( ) = P X Therefore left hand limit of F ( x ) at x is F ( f ( − = P X Similarly for other jump points, we have f ( − = P X . f ( ) = P X , f ( ) = P X . Therefore the probability mass function is − − The distribution function F x ( ) has jumps at x , and .
The jumps are respectively , and . is shown in the figure given below . These jumps determine the probability mass function Probability Distributions Fig. .