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hw4 - ORIE 3500/5500 Engineering Probability and Statistics...

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ORIE 3500/5500 – Engineering Probability and Statistics II Fall 2010 Assignment 4 Problem 1 The joint pmf of X and Y is given by p X,Y ( i, j ) = 1 3 n - 2 for 1 i, j n, | i - j | ≤ 1 , where a positive integer n 4 is a parameter. ( a ) Verify that this is a legitimate joint pmf. ( b ) Compute the marginal probability mass functions. Are X and Y independent? Problem 2 Let the joint density of X and Y be given by f X,Y ( x, y ) = xe - xy if 0 < x < 1 , y > 0 . ( a ) Can you say whether X and Y are independent without comput- ing the marginal densities? ( b ) Now compute the marginal densities and verify your answer to part ( a ). Problem 3 Let X 1 , . . . , X n be independent random variables, each having the standard mean 1 exponential distribution. ( a ) Let M =minimum( X 1 , . . . , X n ). Show that the cdf of M is given by F M ( x ) = 1 - e - nx , x > 0 . Can you name this random variable? ( b ) Compute the cdf and the pdf of M 1 =maximum( X 1 , . . . , X n ). Problem 4 The joint density of X and Y is given by f X,Y ( x, y ) = ( ax + by if 0 < x, y < 1 0 otherwise . If E [( X + 1) Y ] = 11 / 12, find a and b . Problem 5 Each day different meteorologists give us the “proba- bility” that it will rain the next day. To judge how well these people
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