131A_1_ds6_2010fall_Sol

131A_1_ds6_2010fall_Sol - EE 131A Probability Instructor:...

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EE 131A Discussion Set 6 Probability Wednesday, November 1, 2010 Instructor: Lara Dolecek and Friday, November 3, 2010 Reading: Chapters 4 of Probability, Statistics, and Random Processes by A. Leon-Garcia 1. Characteristic function of Gaussian. Problem 4.106, page 225 of ALG Solution : The characteristic function of Y = aX + b is given by Φ Y ( ω ) = E [ e jωY ] = E [ e ( aX + b ) ] = e jωb E [ e jaωX ] = e jωb e jamω - a 2 σ 2 ω 2 / 2 which is the characteristic function of Gaussian random variable with mean of am + b and variance of a 2 σ 2 2. Characteristic function of discrete uniform. Problem 4.114, page 225 of ALG. Solution : (a) The pgf G U ( z ) for the discrete uniform random variable U is given by: G U ( z ) = E [ z U ] = X k =0 P U ( k ) z k = b X k = a 1 b - a + 1 z k = 1 b - a + 1 [ z a - z b +1 1 - z ] (b) The mean and variance of X can be computed from pgf: E ( U ) = G 0 U (1) = lim z 1 1 b - a + 1 1 (1 - z ) 2 [ z a ( az - 1 - a + 1) - z b ( b + 1 - zb )] = lim
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131A_1_ds6_2010fall_Sol - EE 131A Probability Instructor:...

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