## Expected Value of a Random Variable

A probability distribution can have a mean or average value. This value is known as the expected value or the expectation.

The symbol for the expected value of a distribution of x is written E(X).

E(X) = μ (mu, the same symbol as the population mean.)

### Calculation of E(X)

To find the expected value, E(x) of a probability distribution each value of x is multiplied by its probability and the results are added together. This gives E(x).

Using the probability distribution of a six-sided die:

 x 1 2 3 4 5 6 P(X = x)

The expected value E(X) = 1 x  + 2 x  + 3 x  + 4 x  + 5 x  + 6 x  = 3.5

Note The expected value does not have to be one of the values of the distribution. (You cannot score 3.5 on one throw but if you threw the die 100 times and added the scores together you would expect to get around 3.5 × 100 = 350.)

In general:

 E(X) = Σ x . P(X = x)

This formula is in the tables provided for exams but should be memorised.

### Symmetrical Probability Distributions

If the probability distribution is symmetrical, the expected value will be the middle one.

Consider the following probability distribution.

 y 1 2 3 4 5 P(Y = y) 0.1 0.15 0.5 0.15 0.1

The expected value E(Y) is obviously 3.

Check

E(Y) = 1 x 0.1 + 2 x 0.15 + 3 x 0.5 + 4 x 0.15 + 5 x 0.1 = 3

The vertical line graph clarifies this: