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zz-27 - Utah State University ECE 6010 Stochastic Processes...

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Utah State University ECE 6010 Stochastic Processes Homework # 5 Solutions 1. Let X 1 ∼ U (0 , 1) and X 2 ∼ U (0 , 1) (independent). Let Y 1 = - 2 ln X 1 cos(2 πX 2 ) and Y 2 = - 2 ln X 1 sin(2 πX 2 ). Show that Y 1 ∼ N (0 , 1) and Y 2 ∼ N (0 , 1) Here we have, Y 2 1 + Y 2 2 = - 2 ln X 1 [(cos(2 πX 2 )) 2 + (sin(2 πX 2 )) 2 ] = - 2 ln X 1 Therefore, X 1 = e - Y 2 1 + Y 2 2 2 Also, Y 2 /Y 1 = tan(2 πX 2 ), therefore X 2 = 1 2 π tan - 1 ± Y 2 Y 1 ² Jacobian is given by J = " ∂x 1 ∂y 1 ∂x 1 ∂y 2 ∂x 2 ∂y 1 ∂x 2 ∂y 2 # Therefore, | J | - 1 = . . . = X 1 2 π Now, f Y 1 ,Y 2 ( y 1 , y 2 ) = | J | - 1 f X 1 ,X 2 ( x 1 , x 2 ) = X 1 2 π f X 1 ( x 1 ) f X 2 ( x 2 ) = 1 2 π e - Y 2 1 + Y 2 2 2 = 1 2 π e - Y 2 1 2 · 1 2 π e - Y 2 2 2 Y 1 ∼ N (0 , 1) and Y 2 ∼ N (0 , 1) 2. If X and Y are independent and Y ∼ U (0 , 1), show that Z = X + Y has density f Z ( z ) = F x ( z ) - F x ( z - 1). Here

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zz-27 - Utah State University ECE 6010 Stochastic Processes...

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