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131A_1_ds3_2010fall_ZZsol

# 131A_1_ds3_2010fall_ZZsol - EE 131A Probability Instructor...

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EE 131A Discussion Set 3 Probability Wednesday, October 13, 2010 Instructor: Lara Dolecek and Friday, October 15, 2010 Reading: Chapters 3 and 4 of Probability, Statistics, and Random Processes by A. Leon-Garcia 1. Coin tossing example. Problem 3.9, page 131 of ALG Solution :(a) Let the random variable X be the number of heads, then S X = { 0 , 1 , ..., n } . Then the random variable Y , which is difference between the num- ber of heads and the number of tails in the n tosses of a coin, is given by Y = X - ( n - X ) = 2 X - n . Therefore the sample space S Y is given by S Y = {- n, - n + 2 , ...n - 2 , n } (b) P [ Y = 0] = P [2 X - n = 0] = P [ X = n/ 2] = ( n n/ 2 ) p n/ 2 (1 - p ) n/ 2 for n is even (c) P [ Y = k ] = P [2 X - n = k ] = ( n ( n + k ) / 2 ) p ( n + k ) / 2 (1 - p ) ( n - k ) / 2 for n+k is even Problem 3.26, page 133 of ALG Solution : Note Y = 2 X - n and X is the number of heads in n tosses of a coin, which is the binomial random variable. In the class, we already proved E ( X ) = np and V ar ( X ) = np (1 - p ). Therefore E ( Y ) and V ar ( Y ) are given by E ( Y ) = E (2 X - n ) = 2 E ( X ) - n = 2 np - n V ar ( Y ) = V ar (2 X - n ) = 4 V ar ( X ) = 4 np (1 - p )

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131A_1_ds3_2010fall_ZZsol - EE 131A Probability Instructor...

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