Chapter 5 Discrete Probability Distributions 5 3 5

Chapter 5 Discrete Probability Distributions 5. 3 5. 4 EXPECTATION 5. 3. 1 The Mean and Expectation (Expected Value) 5. 3. 2 Some Applications VARIANCE AND STANDARD DEVIATION

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

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

n If we have a statistical experiment: n n 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) =

n Example 1 n random variable X has a probability function defined as shown. Find E(X). P(X= x) -2 -1 0 1 2 0. 3 0. 15 0. 4 0. 05

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

n Example n 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.

n Example n The random variable X has probability function P(X = x) for x = 1, 2, 3. x 1 2 3 P(X = x) 0. 1 0. 6 0. 3 n n Calculate (a) E(3), (b) E(X), (c) E(5 X), (d) E(5 X+3), (e) 5 E(X) + 3, (f) E(X 2), (g) E(4 X 2 - 3), (h) 4 E(X 2 ) – 3. Comment on your answers to parts (d) and (e) and parts (g) and (h).

E(a X + b) = a E(X) + b, where a and b are any constants. E[f 1(X) f 2(X)] = E[f 1(X)] E[f 2(X)], where f 1 and f 2 are functions of X.

n 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