The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics Lecture 13 Means and Variances of Random Variables 5 28 10 Lecture 13 1 Means Variances of Random Variables Data Sample mean x Sample variance s 2 Random variable X Mean E X X 2 Variance Var X X E X and Var X are summaries of the probability distribution of X 5 28 10 Lecture 13 2 Mean of a Discrete Random Variable definition 5 28 10 Lecture 13 3 Free Throws A Tar Heel basketball player is a 80 free throw shooter Suppose he shoots 10 free throws during each practice X number of free throws he makes in a practice Find E X 5 28 10 Lecture 13 4 Free throws continued E X 0 P X 0 1 P X 1 2 P X 2 10 P X 10 by definition Need to compute P X x for x 0 1 10 Any better method Intuition E X 10 0 8 8 In fact it s correct Why Some useful rules create shortcuts detail later 5 28 10 Lecture 13 5 Rules for Expected Value 5 28 10 Lecture 13 6 Variance of a Discrete Random Variable definition 5 28 10 Lecture 13 7 Car Sales The total number X of cars to be sold next week is described by the following probability distribution x p x 0 1 05 15 2 35 3 25 4 20 Find E X and Var X 5 X xi p xi 0 0 05 1 0 15 2 0 35 3 0 25 4 0 20 2 40 i 1 5 X xi 2 4 2 p xi 0 2 4 2 05 1 2 4 2 15 2 i 1 2 2 4 2 35 3 2 4 2 25 4 2 4 2 20 1 24 X 1 24 1 11 5 28 10 Lecture 13 8 5 28 10 Lecture 13 9 Car Salesman s Commission continued Suppose a salesman earns a fixed weekly wage of 150 plus 200 commission for each car sold Let Y be his weekly earnings wage commission Find E Y and Var Y Note Y 150 200 X E Y 150 200 E X 150 200 2 4 630 Var Y 200 2 Var X 40000 1 24 49600 5 28 10 Lecture 13 10 Rules for Variances If X and Y are independent random variables and a and b are constants then 2 aX bY a b 2 2 X 2 2 Y This is a special case 5 28 10 Lecture 13 11 Rules for Variances If X and Y are random variables with correlation then 2 2 2 2 2 aX a b Y 2ab X Y bY X This is a general rule 5 28 10 Lecture 13 12 Women s Height The height of American women aged 18 24 is approximately normally distributed with mean 64 3 inches and s d 2 4 inches Two women in the age group are randomly selected What is the interquartile range of the heights of women in the age group done before What are the mean and the s d of the height difference between the two women 5 28 10 Lecture 13 13 Women s Height continued X Y X Y 64 3 64 3 0 2X Y 2X 2Y 2 42 2 42 11 52 so X Y 3 39 Need to find x1 x2 so that P X x1 25 P X x2 25 25 P X x1 P Z x1 64 3 2 4 From the table P Z 0 67 25 So x1 64 3 2 4 0 67 i e x1 62 69 Similarly 25 P X x2 P Z x2 64 3 2 4 P Z 0 67 25 So x2 64 3 2 4 0 67 i e x2 65 91 So IQR 65 91 62 69 3 22 in 5 28 10 Lecture 13 14 Investment Portfolio Zadie has invested 20 of her funds in Treasury Bills and 80 in an index fund that consists of many U S common stocks Let X be the annual return on T bills and Y the annual return on the index fund Suppose X 5 2 X 2 9 Y 13 3 Y 17 0 0 1 What are the mean and variance of the portfolio return 5 28 10 Lecture 13 15 Investment Portfolio continued Portfolio return R 0 2 X 0 8 Y E R 0 2 E X 0 8 E Y 0 2 5 2 0 8 13 3 Var R 0 2 2 2X 0 8 2 2Y 2 0 2 0 8 corr X Y X Y 0 04 2 9 2 0 64 17 2 2 0 2 0 8 0 1 2 9 17 5 28 10 Lecture 13 16 Take Home Message Definition of mean E X and variance Var X for random variable X How to compute E X and Var X 1 Use definition for discrete RV s 2 Use appropriate rules for discrete and continuous RV s a more flexible and extensively used method 5 28 10 Lecture 13 17
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