180ahw7 - n ) that Y = X 1 + . . . + X n has the negative...

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MATH 180A – Introduction to Probability Homework 7 (due Wedenesday, 11/29/06) 1. Suppose X has the exponential distribution with parameter λ and Y has the ex- ponential distribution with parameter μ . Assume X and Y are independent. Let Z = X/ ( Y + 1). (a) Compute E ( Z ). (b) Compute the PDF of Z . 2. Let X and Y be random variables with PDF f ( x, y ) = x + y, x, y (0 , 1) , and f ( x, y ) = 0 , otherwise . Define Z = X + Y . Compute E ( Z ) and var( Z ). *Bonus* Compute the PDF of Z . 3. Suppose X 1 , X 2 are independent, each with the uniform distribution on (0 , 1). Compute the probability density function of Y = X 1 + X 2 . 4. Show by induction (on
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Unformatted text preview: n ) that Y = X 1 + . . . + X n has the negative binomial distri-bution with parameters ( n, p ) when the X i s are independent, each with the geometric distribution with parameter p . Hint: use the following identity X k = r k r = + 1 r + 1 . 5. Let X 1 , . . . , X n be independent, with X i having the exponential distribution with pa-rameter i . Dene Y = n i =1 X i . Compute E ( Y k ) for k = 1 , . . . , 4. (Doing this for n = 4 will be full credit.)...
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