131A_1_ds5_2010fall_Sol

131A_1_ds5_2010fall_Sol - EE 131A Discussion Set 5...

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Unformatted text preview: EE 131A Discussion Set 5 Probability Wednesday, October 20, 2010 Instructor: Lara Dolecek and Friday, October 27, 2010 Reading: Chapters 4 of Probability, Statistics, and Random Processes by A. Leon-Garcia 1. Max of iid. uniform. Problem 4.174, page 231 of ALG Solution : (a) the random variable Y is given by Y = max { X 1 , X 2 , ..., X n } , then we can compute the cdf of Y as follows: P [ Y y ] = P [max { X 1 , X 2 , ..., X n } y ] = P [ X 1 y, X 2 y, ..., X n y ] = P [ X 1 y ] P [ X 2 y ] ...P [ X n y ] = P [ X y ] n = ( y a ) n (b) Given cdf function in (a), we first compute the pdf function of Y as follows: f Y ( y ) = d F Y ( y ) d y = d( y a ) n d y = ny n- 1 a n then the expectation and variance of Y is given by: E ( Y ) = Z a yf Y ( y )d y = Z a y ny n- 1 a n d y = n n + 1 y n +1 a n | a = n n + 1 a E ( Y 2 ) = Z a y 2 f Y ( y )d y = Z a y 2 ny n- 1 a n d y = n n + 2 y n +2 a n | a = n n + 2 a 2 V ar ( Y ) = E ( Y 2 )- E ( Y ) 2 = n n + 2 a 2...
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131A_1_ds5_2010fall_Sol - EE 131A Discussion Set 5...

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