Unformatted text preview: Y = X 1 X 1 + X 2 , the proportion of time that the machine is in operation during any one operationrepair 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 ( n1)! eλy y n1 , y ≥ Hence, argue that Y follows Gamma( λ,n ) using induction on n . 1...
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 Summer '08
 WEBER
 Normal Distribution, Variance, Probability theory, probability density function, joint density

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