HW4soln

# HW4soln - OR 3500/5500 Summer’08 Homework 4 Homework 4...

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Unformatted text preview: OR 3500/5500, Summer’08, Homework 4 Homework 4 Due on Thursday, June 5, 11am. Problem 1 Suppose that density of X is given by f ( x ) = a + bx 2 , ≤ x ≤ 1 If E(X)=c, compute a and b in terms of c . Solution Since R f ( x ) dx = 1, a + b 3 = 1 (1) We are given E ( X ) = R xf ( x ) dx = c , from which it follows that a 2 + b 4 = c (2) Solving for a and b from (1) and (2), a = 3- 4 c b = 12 c- 6 Problem 2 Let X 1 ,X 2 ,...,X n be independent Uniform(0,1) random variables ( ie , each one of them have density given by f ( x ) = 1 , ≤ x ≤ 1). (a) Compute the cdf of Y := min( X 1 ,...,X n ). (b) Use (a) to compute the density of Y . (c) Find E ( Y ). Solution (a) Clearly Y takes values in [0 , 1]. Let 0 ≤ y ≤ 1. F Y ( y ) = 1- P ( Y > y ) = 1- P ( X i > y for 1 ≤ i ≤ n ) = 1- Π n i =1 P ( X i > y ) = 1- (1- y ) n Thus, cdf of Y is F Y ( y ) = , y < 1- (1- y ) n , o ≤ y ≤ 1 1 , y > 1 (b) Differentiating the cdf, we get that the density of Y is f y ( y ) = n (1- y ) n- 1 , ≤ y ≤ 1 1 OR 3500/5500, Summer’08, Homework 4 (c) E ( Y ) = Z ∞ { 1- F Y ( y ) } dy = Z 1 (1- y ) n dy =- (1...
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HW4soln - OR 3500/5500 Summer’08 Homework 4 Homework 4...

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