Lesson 6 0 Mean and Variance Transformations Objectives

Lesson 6 - 0 Mean and Variance Transformations

Objectives • Understand how linear combinations affect the mean of a random variable • Understand how linear combinations affect the variance of a random variable

Vocabulary • E(x) – the expected value of a random variable, x; also known as the mean • V(x) – the variance of a random variable, x

Rules for Means • The expected value of a random variable is the mean E(X) = μx (note population parameter notation) • Rule 1: If X is a random variable and a and b are constants, then E(a. X + b) = aμx + b • Rule 2: If X and Y are random variables, then E(X + Y) = E(X) + E(Y) = μx + μy E(X – Y) = E(X) – E(Y) = μx – μy

Rules for Variances • The variance of a random variable is V(X) = σ²x (note population parameter notation) • Rule 1: If X is a random variable and a and b are constants, then V(a. X + b) = a²σ²x (note addition does not affect V(X)) • Rule 2: If X and Y are independent random variables, then V(X + Y) = V(X) + V(Y) = σ²x + σ²y V(X – Y) = V(X) + V(Y) = σ²x + σ²y

Example Problem The probability of scoring X touchdowns in a game for the Scarlet Hurricanes is given by the following: Touchdowns Probability 0 1 2 0. 05 0. 25 0. 35 3 0. 2 4 0. 15 The probability of scoring Y field goals in a game is given by the following: Field Goals Probability 0 0. 4 1 0. 4 2 3 0. . 15 0. 05 4 0. 0 How many points per game, if all PATs are good, do the Scarlet Hurricanes score and what is their variance? Score, S = 7 X + 3 Y

Example Problem Cont To determine the mean and the variance when need to enter the touchdowns in L 1 and their associated probabilities in L 2 Touchdowns Probability 0 1 2 3 4 0. 05 0. 25 0. 35 0. 2 0. 15 We get μx = 2. 15 and σx = 1. 107926. V(X) = 1. 2275 Now repeat the procedure for field goals Field Goals Probability 0 1 2 3 4 0. . 15 0. 0 We get μy = 0. 85 and σy = 0. 852936. V(Y) = 0. 68784

Example Problem Cont • Since the Score, S = 7 X + 3 Y, we now can use our rules to figure out E(S) and V(S). We got μx = 2. 15 and σx = 1. 107926. V(X) = 1. 2275 We got μy = 0. 85 and σy = 0. 852936. V(Y) = 0. 68784 So E(S) = 7 E(X) + 3 E(Y) = 7μx + 3μy = 7(2. 15) + 3(0. 85) = 17. 6 pts/game (average) So V(S) = 7²V(X) + 3²V(Y) = 49 V(X) + 9 V(Y) = 49(1. 2275) + 3(0. 68784) = 62. 21102 pts/game (variance)

Summary and Homework • Summary – Adding a constant to a random variable moves the mean, but does not change the variance – Multiplying a random variable by a constant shifts the mean by the constant multiple and shifts the variance by the square of the constant • Homework: None