The Expected Value of A Discrete Random Variable

The expected value of a discrete random variable of a probability distribution is the mean of the random variable of the distribution. We write E(X) for expected value of the random variable X. The formula is

Lets revisit the newsboy problem and calculate the expected value for the number of papers sold.

x	p(x)
0	0.10
1	0.12
2	0.35
3	0.20
4	0.15
5	0.08

First we begin by creating a new column x p(x).

x	p(x)	x px)
0	0.10	0.0
1	0.12	0.12
2	0.35	0.7
3	0.20	0.60
4	0.15	0.60
5	0.08	0.40

Next sum the numbers in the column x p(x)

The expected value is: E(x)=2.42
What does this mean? On some days the boy will sell 2 papers, on other days he will sell 4 papers etc. Theoretically If we keep track of the number of papers sold over millions and millions of days we would find that the average number of papers sold would be very close to 2.42. Sometimes the expected value of a random variable is thought of as the long term mean. The term expected value is used in the gamming industry, insurance and decision theory. We use expected value rather than mean whenever probabilities are involved. However, often you will see the familar x-bar used instead of E(x)