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AP STATISTICS Chapter 7 – Random Variables Section 7.2: Means and Variances of Random Variables (1) Homework Assignment: Exercises 7.22,7.23,7.29 Name _______________________ Date __________ Period _____ A. Review Assignment: Exercises 7.2, 7.6, 7.8, 7.10, 7.12, 7.16 B. Key Vocabulary: mean of a random variable ( x ), variance of a random variable ( X2 ), expected value of X [ E(X) ], X xi pi , X2 xi X pi , standard deviation of X, ( X ), law of large numbers, 2 rules for means: a bX a b X , X Y X Y , rules for variances: a2bX b 2 X2 , X2 Y X2 Y X2 Y2 (so long as X and Y are independent – variances add), linear combination of independent normal random variables is also normally distributed, correlation between two independent random variables is zero Whoa!!!!! Some pretty intimidating looking stuff up there!! C. Discrete Random Variable Situation: Finding mean (expected value), variance, and standard deviation Let X be the number of heads counted when 3 coins are tossed. X is a discrete random variable. X P(X) 0 1/8 1 3/8 2 3/8 3 1/8 Sketch the probability histogram: 0 1 2 3 Mean (expected value) calculation: X xi pi Variance calculation: X2 xi X pi 2 Standard deviation calculation (square root of variance): (Do on calculator: L1 values of X, L2 values of P(X); 1-Var Stats L1,L2) D. Continuous Random Variable Situations The mean and variance (or standard deviation) are provided for you in the question. Page 422: Example 7.12 SAT Scores SAT Math score (X) SAT Verbal score (Y) X = 625 Y = 590 X = 90 Y = 100 E. The Law of Large Numbers Draw independent observations at random from any population with finite mean . Decide how accurately you would like to estimate . As the number of observations drawn increases, the mean x of the observed values eventually approaches the mean of the population as closely as you specified and then stays that close. Example: The distribution of the heights of all young women is close to N(64.5,2.5). Pick out how close you would like your sample mean to be to the truth (64.5). As the sample size n is increased, x Calculus students: Limits notation, anyone? F. Rules for Means and Variances: If the transformation a+bx is applied to the values of a random variable, then a bX a b X and a2bX b 2 X2 Example: Suppose the temperatures (F) of patients in a certain surgical situation are controlled to be N(98.1,.2). What would the mean ( C ) and variance ( C2 ) be if we transform by converting to Celsius? F = (9/5)C + 32 C = -17.78 +(5/9)F C 17.78(5 / 9) F 17.78 (5 / 9) F 17.78 (5 / 9)(98.1) 36.72 C2 217.78(5 / 9) F (5 / 9) 2 F2 (5 / 9) 2 .2 2 0.01234 and C 0.01234 0.1111