s7 - Math 4653: Elementary Probability: Spring 2007...

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Unformatted text preview: Math 4653: Elementary Probability: Spring 2007 Homework #7. Problems and Solutions 1. Ch. 4, Review: #21: Suppose R 1 and R 2 are two independent random variables with the same density function f ( x ) = x exp(- 1 2 x 2 ) for x 0. Find a) the density of Y = min { R 1 ,R 2 } ; b) the density of Y 2 ; c) E ( Y 2 ). Solution. a) For each y > 0, we have P ( R k > y ) = Z y f ( x ) dx = exp- 1 2 y 2 , k = 1 , 2; P ( Y > y ) = P (min { R 1 ,R 2 } > y ) = P ( R 1 > y, R 2 > y ) = P ( R 1 > y ) P ( R 2 > y ) = exp(- y 2 ) . Therefore, f Y ( y ) =- d dy P ( Y > y ) = 2 y exp(- y 2 ) for y > , and f Y ( y ) = 0 for y 0. b) The density of Z = Y 2 is f Z ( z ) = f Y ( y ) dz/dy = 2 y exp(- y 2 ) 2 y = e- z for z > , and f Z ( z ) = 0 for z 0. c) E ( Y 2 ) = E ( Z ) = = 1. 2. Sec. 5.2: #16: Suppose X 1 ,X 2 ,X 3 are independent exponential random variables with parameters 1 , 2 , 3 respectively. Evaluate P ( X 1 < X 2 < X 3 ). Solution. P ( X 1 < X 2 < X 3 ) = ZZZ x 1 <x 2 <x 3 1 e- 1 x 1 2 e- 2 x 2 3 e- 3 x 3 dx 1 dx 2 dx 3 = Z 1 e- 1 x 1 " Z x 1 2 e- 2 x 2 Z x 2 3 e- 3 x 3 dx 3 ! dx 2 # dx 1 = Z 1 e- 1 x 1 " Z x 1 2 e- ( 2 + 3 ) x 2 dx 2 # dx 1 = 1 2 2 + 3 Z e- ( 1 + 2 + 3 ) x 1 dx 1 = 1 2 ( 2 + 3 )( 1 + 2 + 3 ) . 3. Sec. 5.3: #2: Let X and Y be independent random variables, with E ( X ) = 1, E ( Y ) = 2, Var ( X ) = 3, and Var ( Y ) = 4. a) Find E (10 X 2 + 8 Y 2- XY + 8 X + 5 Y- 1)....
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This note was uploaded on 09/14/2009 for the course STAT 134 taught by Professor Aldous during the Fall '03 term at University of California, Berkeley.

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s7 - Math 4653: Elementary Probability: Spring 2007...

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