Derived Relationship Between Probability Function and Cumulative Probability Function

Let X be a random variable with probability function P(x) and cumulative probability function
F(). Then we can show that

where the notation implies that summation is over all possible values of x that are less than or equal to.

 Derived Properties of Cumulative Probability Functions for Discrete Random Variables
Let X be a discrete random variable with cumulative probability function F(). Then we can show that
 

Properties of Discrete Random Variables

(a) Expected Value of a Discrete Random Variable
The expected value, E(X), of a discrete random variable X is defined as

                            

where the notation indicates that summation extends over all possible values of x.

The expected value of a random variable is also called its mean and is denoted  .

Example: Suppose that the probability function for the number of errors, X, on pages from business textbooks is

                P(0) = 0.81       P(1) = 0.17         P(2) = 0.02

Find the mean number of errors per page.

Solution: We have

                
From this result it is concluded that over a large number of pages, the expectation would be to find an average of 0.21 error per page.

The concept of variance can be very useful in comparing the dispersions of probability distributions. Consider, for example, viewing as a random variable the return over a year on an investment. Two investments may have the same expected returns but will still differ in an important way if the variances of these returns are substantially different from the mean are more likely than if the variance of returns is small. In this context, then, variance of the return can be associated with the concept of the risk of an investment—the higher the variance, the greater the risk.

Taking the square root of the variance to obtain the standard deviation yields a quantity in the original units of measurement.

Example: An automobile dealer calculates the proportion of new cars sold that have been returned various numbers of times for the correction of defects during the warranty period. The results are shown in the table.

Number of returns	0	1	2	3	4
Proportion	0.28	0.36	0.23	0.09	0.04

(a)    Find the mean number of returns of an automobile for the corrections for defects during the warranty period.
(b)    Find the variance of the number of returns of an automobile for corrections for defects during the warranty period.