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# 09hw8 - EE 464 Mitra Homework 8 Due Friday 9am 464 Box...

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EE 464, Mitra Spring 2009 Homework 8 Due Friday, March 27, 2009, 9am 464 Box This problem set investigates tail inequalities, transforms and preliminaries for bivariate random variables. 1. Bivariate Gaussian random variables (a) Show that the probability density function provided in class for two jointly Gaussian random variables [ X, Y ] T is equivalent to the following expression: p Z ( z ) = 1 (2 π ) n 2 p | Σ | exp - 1 2 ( z - m ) T Σ - 1 ( z - m ) where Z = [ X, Y ] T m = [ m X , m Y ] T Σ = σ 2 X ρσ X σ Y ρσ X σ Y σ 2 Y and n = 2 Note that the notation | · | denotes taking the determinant. (b) The Matlab command x = randn(2,2000) will generate 2000 samples of a bivariate Gaus- sian random variable X = [ X 1 , X 2 ] T with mean vector and covariance matrix m = 0 0 Σ = 1 0 0 1 We will soon learn that we can convert these two independent, standard Gaussian random variables into two Gaussians with arbitrary mean and covariance matrix through the following linear transformation (note that I is the two-by-two identity matrix): X N (0 , I ) Y = AX + m Y N ( m , AA T )

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09hw8 - EE 464 Mitra Homework 8 Due Friday 9am 464 Box...

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