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Unformatted text preview: OR 3500/5500, Summer’08, Homework 3 Homework 3 Due on Monday, June 2, 4pm. Problem 1 Suppose that X is a continuous random variable with the following density f ( x ) = 1 / 3 , 2 ≤ x ≤ 1 Compute the density of X 2 . Solution We first compute the cdf of X 2 and use that to compute the density. Let ≤ x ≤ 1 P ( X 2 ≤ x ) = P ( √ x ≤ X ≤ √ x ) = Z √ x √ x f ( t ) dt = 2 3 √ x Now suppose 1 < x ≤ 4 P ( X 2 ≤ x ) = P ( √ x ≤ X ≤ √ x ) = Z √ x √ x f ( t ) dt = Z 1 √ x 1 / 3 dt = 1 3 (1 + √ x ) Thus the cdf of X 2 is P ( X 2 ≤ x ) = , x < 2 3 √ x, ≤ x ≤ 1 1 3 (1 + √ x ) , 1 < x ≤ 4 1 , x > 4 Differentiating the above, we get that the density of X 2 is f X 2 ( x ) = 1 3 x 1 / 2 , ≤ x ≤ 1 1 6 x 1 / 2 , 1 < x ≤ 4 , otherwise Problem 2 A professor asks her student to do a certain experiment and report some mea surement. The measurement is a number between 0 and 1 with the following pdf: f ( x ) = cx, ≤ x < 1 / 2 c (1 x ) , 1...
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This note was uploaded on 06/15/2008 for the course ORIE 360 taught by Professor Ehrlichman during the Summer '07 term at Cornell.
 Summer '07
 EHRLICHMAN

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