Required Properties of Probability Distribution

Let X be a discrete random variable with probability distribution function P(x). Then

Cumulative Probability Function

The cumulative probability function, F(Xo), for a random variable X, expresses the probability that X does not exceed the value Xo, as a function of Xo. That is, where the function is evaluated at all values of Xo.
Example: Stetson Motors, Inc., is a car dealer in a small Midwestern town. Based on an analysis of its sales history, the managers know that on any single day the number of Vertigo A cars sold can vary from 0 to 5. How can the probability distribution function shown in the following table be used for inventory planning.

x	P(x)	F(x)
0	0.15	0.15
1	0.30	0.45
2	0.20	0.65
3	0.20	0.85
4	0.10	0.95
5	0.10	0.95
6	0.05	1.00

Solution: The random variable, X, takes on the values of x indicated in the first column, and the probability function, P(x), is defined in the second column. The third column contains the cumulative distribution, F(x). This model could be used for planning the inventory of cars. For example, if there are only four cars in stock, Stetson Motors could satisfy customers’ needs for a car 95% of the time. But if only two cars are in stock, then 35% [(1 –0.65)x100%] of the customers would not have their needs satisfied.