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Problem Solutions – Chapter 6 Problem 6.1.1 Solution The random variable X 33 is a Bernoulli random variable that indicates the result of flip 33. The PMF of X 33 is P X 33 ( x ) = 1 px = 0 = 1 0 otherwise (1) Note that each X i has expected value E [ X ]= p and variance Var [ X p ( 1 p ) . The random variable Y = X 1 +···+ X 100 is the number of heads in 100 coin flips. Hence, Y has the binomial PMF P Y ( y ) = ( ± 100 y ) p y ( 1 p ) 100 y y = 0 , 1 ,..., 100 0 otherwise (2) Since the X i are independent, by Theorems 6.1 and 6.3, the mean and variance of Y are E [ Y ] = 100 E [ X ] = 100 p Var [ Y 100 Var [ X 100 p ( 1 p ) (3) Problem 6.1.2 Solution Let Y = X 1 X 2 . (a) Since Y = X 1 + ( X 2 ) , Theorem 6.1 says that the expected value of the difference is E [ Y ] = E [ X 1 ] + E [ X 2 ] = E [ X ] E [ X ] = 0 (1) (b) By Theorem 6.2, the variance of the difference is [ Y [ X 1 ]+ [− X 2 2Var [ X ] (2) Problem 6.1.3 Solution (a) The PMF of N 1 , the number of phone calls needed to obtain the correct answer, can be determined by observing that if the correct answer is given on the n th call, then the previous n 1 calls must have given wrong answers so that P N 1 ( n ) = ² ( 3 / 4 ) n 1 ( 1 / 4 ) n = 1 , 2 ,... 0 otherwise (1) (b) N 1 is a geometric random variable with parameter p = 1 / 4. In Theorem 2.5, the mean of a geometric random variable is found to be 1 / p . For our case, E [ N 1 4. (c) Using the same logic as in part (a) we recognize that in order for n to be the fourth correct answer, that the previous n 1 calls must have contained exactly 3 correct answers and that the fourth correct answer arrived on the n -th call. This is described by a Pascal random variable. P N 4 ( n 4 ) = ² ± n 1 3 ) ( 3 / 4 ) n 4 ( 1 / 4 ) 4 n = 4 , 5 0 otherwise (2) 243

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(d) Using the hint given in the problem statement we can find the mean of N 4 by summing up the means of the 4 identically distributed geometric random variables each with mean 4. This gives E [ N 4 ]= 4 E [ N 1 16. Problem 6.1.4 Solution We can solve this problem using Theorem 6.2 which says that Var [ W [ X ]+ [ Y 2Cov [ X , Y ] (1) The first two moments of X are E [ X ] = Z 1 0 Z 1 x 0 2 xdydx = Z 1 0 2 x ( 1 x ) dx = 1 / 3 (2) E ± X 2 = Z 1 0 Z 1 x 0 2 x 2 dydx = Z 1 0 2 x 2 ( 1 x ) = 1 / 6 (3) (4) Thus the variance of X is Var [ X E [ X 2 ]− ( E [ X ] ) 2 = 1 / 18. By symmetry, it should be apparent that E [ Y E [ X 1 / 3 and Var [ Y [ X 1 / 18. To find the covariance, we first find the correlation E [ XY ] = Z 1 0 Z 1 x 0 2 xydydx = Z 1 0 x ( 1 x ) 2 = 1 / 12 (5) The covariance is Cov [ X , Y ] = E [ ] E [ X ] E [ Y ] = 1 / 12 ( 1 / 3 ) 2 =− 1 / 36 (6) Finally, the variance of the sum W = X + Y is [ W [ X [ Y X , Y ] = 2 / 18 2 / 36 = 1 / 18 (7) For this specific problem, it’s arguable whether it would easier to find Var [ W ] by first deriving the CDF and PDF of W . In particular, for 0 w 1, F W (w) = P [ X + Y w ] = Z w 0 Z w x 0 2 = Z w 0 2 (w x ) = w 2 (8) Hence, by taking the derivative of the CDF, the PDF of W is f W (w) = ² 2 w 0 w 1 0 otherwise (9) From the PDF, the first and second moments of W are E [ W ] = Z 1 0 2 w 2 d w = 2 / 3 E ± W 2 = Z 1 0 2 w 3 d w = 1 / 2 (10) The variance of W is Var [ W E [ W 2 ( E [ W ] ) 2 = 1 / 18. Not surprisingly, we get the same answer both ways.
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