Fall 2010 HW12 - University of Illinois Fall 2010 ECE 313...

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University of Illinois Fall 2010 ECE 313: Problem Set 12: Solutions Joint pdfs, Covariance, LLN and CLT 1. [Joint pdfs of functions of random variables] (a) The joint pdf of X and Y is given by: f X,Y ( u,v ) = ( 1 (1 - 2 Q (3)) 2 π (0 . 01) e - ( v - 0 . 3) 2 0 . 02 1 (0 . 4) 0 . 8 V u 1 . 2 V, 0 v 0 . 6 V 0 else where 1 - 2 Q (3) is the probability a N (0 . 3 , 0 . 01) random variable falls in the interval [0 . 0 . 6] . Since Q (3) = 0 . 0013 , the factor 1 1 - 2 Q (3) is approximately 1.0026, which is so close to one that the distribution of Y is nearly Gaussian. (b) Given: α = k 1 u ( u - v ) β = k 2 u u - v We first solve for v in terms of u from the second equation above to obtain v = ± 1 - k 2 β ² u Then, we substitute for v in the first equation to obtain α = k 1 u 2 - k 1 u ± 1 - k 2 β ² u = k 1 k 2 β u 2 Keeping in the mind that V dd and V t are non-negative, we get u = r αβ k 1 k 2 v = ± 1 - k 2 β ² r αβ k 1 k 2 Thus, ( W,Z ) = g ( X,Y ) is an invertible mapping because it is possible to obtain unique values of
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Fall 2010 HW12 - University of Illinois Fall 2010 ECE 313...

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