Chapter 5 Discrete Probability Distributions 5.3 EXPECTATION 5.3.1 The Mean and Expectation (Expected Value) 5.3.2 Some Applications 5.4 VARIANCE AND STANDARD DEVIATION

5.3 EXPECTATION 5.3.1 The Mean and Expectation (Expected Value) Experimental approach Suppose we throw an unbiased die 120 times and record the results: Then we can calculate the mean score obtained where Scroe, x 1 2 3 4 5 6 Frequency, f 15 22 23 19 18 = = = ________ (3 d.p.)

Theoretical approach The probability distribution for the random variable X where X is ‘the number on the die’ is as shown: We can obtain a value for the ‘expected mean’ by multiplying each score by its corresponding probability and summing, so that Expected mean = = Score, x 1 2 3 4 5 6 P(X = x) 1/6

If we have a statistical experiment: a practical approach results in a frequency distribution and a mean value, a theoretical approach results in a probability distribution and an expected value. The expectation of X (or expected value), written E(X) is given by E(X) =

Example 1 random variable X has a probability function defined as shown. Find E(X). -2 -1 1 2 P(X= x) 0.3 0.1 0.15 0.4 0.05

In general, if g(X) is any function of the discrete random variable X then E[g(X)] =

Example In a game a turn consists of a tetrahedral die being thrown three times. The faces on the die are marked 1,2,3,4 and the number on which the die falls is noted. A man wins $ whenever x fours occur in a turn. Find his average win per turn.

Example The random variable X has probability function P(X = x) for x = 1,2,3. Calculate (a) E(3), (b) E(X), (c) E(5X), (d) E(5X+3), (e) 5E(X) + 3, (f) E(X2), (g) E(4X2- 3), (h) 4E(X2 ) – 3. Comment on your answers to parts (d) and (e) and parts (g) and (h). x 1 2 3 P(X = x) 0.1 0.6 0.3

E(a X + b) = a E(X) + b, where a and b are any constants. E[f1(X)  f2(X)] = E[f1(X)]  E[f2(X)], where f1 and f2 are functions of X.

5.3.2 Some Applications

5.4 VARIANCE AND STANDARD DEVIATION The variance of X, written Var(X), is given by Var(X) = E(X - )2