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MEAN & VARIANCE OF RANDOM VARIABLES Chapter 7 EXPECTED VALUE The mean (center) of a discrete random variable, X. It is a weighted average -Each value of X is weighted by its probability. FINDING EXPECTED VALUE To Find: x x1 p1 x2 p2 ...xk pk xi pi Or in other words: Multiply each value of X by its probability and add products This terminology is often used in relation to game of chance or business prediction. USING EXPECTED VALUE Example: Matthew is a doorman. The following table gives the probabilities that customers will give various amounts of money: Amounts of $ 30 35 40 45 50 55 60 Probability .45 .25 .12 .08 .05 .03 .02 What size tip can Matthew expect to get? EXAMPLE 2 Concessionaires know that attendance at a football stadium will be 60,000 on a clear day, 45,000 if there is light snow and 1500 if there is heavy snow. Furthermore the probabilities of clear skies, light snow or heavy snow on a particular day is 1/2, 1/3, and 1/6 respectively. What average attendance should be expected for the game? EXAMPLE 3 10,000 raffle tickets are sold at $2 each for four prizes of $500, $1000, $1500, and $5000. If you buy one ticket what is the expected value of your gain? LAW OF LARGE NUMBERS As the number of observations increases, the mean of the observed values, x , approaches the mean of the population, µ. The more variation in the outcomes, the more trials are needed to ensure that x is close to µ . RULES FOR MEANS If X is a random variable and a and b are fixed numbers, then abx a bx If X and Y are random variables, then X Y X Y We use these for conversions!!! EXAMPLE The equation Y = 20 + 10X converts a PSAT math score, X, into an SAT math score, Y. The average PSAT math score is 48. What is the average SAT math score? EXAMPLE Let X 625 represent the average SAT math score. Let Y 590 represent the average SAT verbal score. What is the average combined score? VARIANCE OF A DISCRETE RANDOM VARIABLE If X is a discrete random variable with mean , then the variance of X is x1 X p1 x2 X p2 ... x k X pk 2 X 2 2 2 x i X 2 pi The standard deviation X is the square root of the variance. RULES FOR VARIANCE If X is a random variable and a and b are fixed numbers, then 2 a bX b 2 2 X If X and Y are independent random variables, then 2 2 2 X Y X Y 2 X Y 2 X 2 Y EXAMPLE The equation Y = 20 + 10X converts a PSAT math score, X, into an SAT math score, Y. The standard deviation for the PSAT math score is 1.5 points. What is the standard deviation for the SAT math score? EXAMPLE The standard deviation for the SAT math score is 150 points, and the standard deviation for the SAT verbal score is 165 points. What is the standard deviation for the combined SAT score? EXAMPLE X and Y are independent random variables. If x 2.3 and y 4.1, find x y . ONE MORE… As head of inventory for Knowway computer company, you were thrilled that you had managed to ship 2 computers to your biggest client the day the order arrived. You are horrified, though, to find out that someone restocked refurbished computers in with the new computers in your storeroom. The shipped computers were selected randomly from 15 computers is stock, but 4 of those were actually refurbished. If your client gets 2 new computers, things are fine. If the client gets a refurbished computer, it will be sent back at your expense - $100- and you can replace it. However, if both computers are refurbished, the client will cancel the order for this month and you will lose $1000. What is the expected value and the standard deviation of your loss?