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# zz-14 - | Z ∞-∞ f XY z-by a y dy Hint Let W = Y be an...

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Utah State University ECE 6010 Stochastic Processes Homework # 5 Due Friday October 8, 2005 1. Box Muller: Let X 1 ∼ U (0 , 1) and X 2 ∼ U (0 , 1) (independent). Let Y 1 = p - 2 ln X 1 cos 2 πX 2 Y 2 = p - 2 ln X 1 sin 2 πX 2 Show that Y 1 ∼ N (0 , 1) Y 2 ∼ N (0 , 1) . 2. If X and Y are independent and Y ∼ U (0 , 1), show that Z = X + Y has the density f Z ( z ) = F x ( z ) - F x ( z - 1) . 3. Let X ∼ N (0 , σ 2 ) and Y ∼ N (0 , σ 2 ) be independent. Let Z = X 2 + Y 2 . Show that f Z ( z ) = 1 2 σ 2 e - z/ 2 σ 2 u ( z ) Such a random variable is said to be chi-squared ( χ 2 ) distributed with two degrees of freedom. 4. If X ∈ U (0 , 1) and Y = - (ln( X )) , show that Y is exponentially distributed. 5. Let g ( x ) be a monotone increasing function and let Y = g ( X ). Show that F XY ( x, y ) = ( F X ( x ) if y > g ( x ) F Y ( y ) if y < g ( x ) . 6. Let Z = aX + bY . Show that f Z ( z ) =

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Unformatted text preview: | Z ∞-∞ f XY ( z-by a , y ) dy. Hint: Let W = Y be an auxiliary variable. 1 7. Let X and Y be independent with f X ( x ) = αe-αx u ( x ) f Y ( y ) = βe-βy u ( y ) . Determine the density of: (a) Z = X/Y . (b) Z = max( X, Y ) . 8. Show that if X, Y are i.i.d. N (0 , σ 2 ), then Z = X/Y has a Cauchy distribution, f Z ( z ) = 1 /π z 2 + 1 . Problems from Grimmet & Stirzaker. 1. Exercise 4.4.3. 2. Exercise 4.7.3. 3. Exercise 4.7.4. 4. Exercise 4.8.1 (Hint: Laplace transform) 5. Exercise 4.8.2 (Hint: Laplace transform) 2...
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