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Unformatted text preview: X and Y be given by f X,Y ( x,y ) = ± ex ≤ y ≤ x < ∞ otherwise 1 Figure 1: Problem 6. Deﬁne Z = X + Y , W = XY . Find the joint pdf of Z and W . Show that Z is not an exponential random variable. Problem 8. Let f X,Y ( x,y ) = ± 2 e( x + y ) < x < y < ∞ otherwise Deﬁne Z = X + Y , W = Y/X . Determine the joint pdf of Z and W . Problem 9. (a) Use the auxiliary variable method to ﬁnd the pdf of Z = X X + Y . (b) Find the pdf of Z if X and Y are independent exponential random variables with the same parameter α . Problem 10. Suppose X and Y are zeromean independent Gaussian random variables with common variance σ 2 . Deﬁne R = √ X 2 + Y 2 and θ = tan1 ( Y/X ), where  θ  < π . Obtain their joint density function. 2...
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 Fall '08
 DASILVER
 Normal Distribution, Variance, Probability theory, probability density function

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