Lab5WithSol - Statistics 103 Lab 5(a P(X > 9 LAB PROBLEMS(b...

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Statistics 103: Lab 5 LAB PROBLEMS 1. Suppose X and Y are the random variables with the joint PMF: X \ Y 1 2 3 1 0.1 0.3 0.1 2 0.2 0 0.1 3 0 0.1 0.1 (a) Compute cov ( X, Y ). (b) Compute var ( X ) (c) Compute var ( Y ) (d) Compute var ( X + Y ) using the variance for- mula. answer: 0 . 1 ; 0 . 61 ; 0 . 6 , 1 . 41 2. Suppose that Z 0 , Z 1 , Z 2 , . . . are a sequence of in- dependent normal random variables with mean 0 and variance σ 2 . Let a stock price on day t be de- scribed by X t = a + bZ t + cZ t - 1 , t = 1 , 2 , 3 , . . . , where a, b , and c are all positive constants. Find Var( X t ), Cov( X t , X t +1 ), and Cov( X t , X t +2 ). From your calculations, does knowing X t give you insight into what X t +1 or X t +2 might be? answer: Var ( X t ) = ( b 2 + c 2 ) σ 2 ; Cov( X t , X t +1 ) = bcσ 2 ; Cov( X t , X t +2 ) = 0; knowing X t should help us predict X t +1 , but not X t +2 . 3. Consider randomly picking a married couple from Minnesota. Let W denoted the height of the wife (in inches) and let H denoted the height of the husband (in inches). Pearson and Lee published the following approximations for the expected values, standard deviations and correlation between H and W : EW = 63 , sd ( W ) = 2 . 5 (1) EH = 68 , sd ( H ) = 2 . 7 (2) ρ = 0 . 25 (3) (a) If W = 66 what do you predict the husbands height to be? (b) What percentage of married women from Min- nesota are over 5 feet 8 inches tall? (c) Of the women who are married to men 6 feet tall, what percentage are over 5 feet 8 inches tall? answer: 68.81; 2.28%; 4.65% 4. Suppose X 1 , ...X 15 is random sample (with replace- ment) from the population with mean 15 and stan- dard deviation 10. Please approximate the follow- ing probabilities using the Central Limit Theorem. (a) P ( X > 9) (b) P (13 . 5 < X < 20 . 1) answer: (a) 0.9898 , (b) 0.6952 5. Let X and Z be two independent standard normal random variables. Define Y to be aX + b + Z . Compute the correlation between Y and X . If you observe that X = 1 . 28 what do you predict for Y ? Does this prediction make sense to you? answer: a a 2 +1 ; a(1.28) + b; hopefully. PRACTICE PROBLEMS 1. In Major League Baseball, two commonly used metrics for the evaluation of pitching are ERA (earned run average) and WHIP (walks plus hits per inning pitched). A scatterplot of all 30 teams’ ERA and WHIP is shown in Figure 1. Use the scat- terplot to estimate the standard deviations of ERA and WHIP and the correlation between ERA and WHIP. Use these values to estimate the covariance between ERA and WHIP. Also, predict the WHIP of a team with an ERA of 4.5. Now, suppose that the variances of ERA and WHIP are 0.5 and 0.01, respectively, and the correlation between the two metrics is 0.9. Calculate the co- variance between ERA and WHIP and compare it to your previous estimate. answer: 0.0636 2. After he gets out of school each day, Calvin spends his remaining waking hours doing his homework, watching TV, or playing Calvinball with Hobbes. Suppose that Calvin gets out of school at 3:00pm every day and goes to sleep at 9:00pm. Let X be the number of hours that Calvin spends doing home- work on a given night, and let Y be the number of hours he spends playing Calvinball. If Calvin’s parents force him to do homework for exactly two hours every night, what is the correlation between X and Y ?
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