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HW9 - Y = X 1 X 1 X 2 the proportion of time that the...

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OR 3500/5500, Summer’09, Chen Homework 9 Due on Monday, June 22, 3pm. Problem 1 Find the joint density of ( U, V ) = ( X + Y, X - Y ), where X and Y are inde- pendent standard normal random variables. Are U and V independent? Problem 2 A machine produces cylindrical containers with the radii and the heights varying according to a joint pdf f R,H ( r, h ) = 2 r ( r + 2) h r +1 if 0 < r, h < 1 0 otherwise . Find the joint density for the volume and surface area of the containers (recall that V = πR 2 H and S = 2 πRH .) Problem 3 The length of time that a certain machine operates without failure is denoted by X 1 and the length of the repair time by X 2 . After repair the machine is assumed to operate like new. Assume that X 1 and X 2 are independent standard exponential random vari- ables. Find the pdf of
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Unformatted text preview: Y = X 1 X 1 + X 2 , the proportion of time that the machine is in operation during any one operation-repair cycle. Problem 4 Let X and Y be independent random variables, X a standard uniform, and Y a standard exponential random variable. Find the density of the sum Z = X + Y . Caution : two diﬀerent cases come up here. The following problem is optional for 3500 students, mandatory for students enrolled in 5500 Problem 5 If X 1 ,X 2 ,...,X n are independent Exp( λ ), show that for n ≥ 2, the density of Y := X 1 + ... + X n is given by f Y ( y ) = λ n ( n-1)! e-λy y n-1 , y ≥ Hence, argue that Y follows Gamma( λ,n ) using induction on n . 1...
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