a3su112131 (1) - In a five-game playo± series between...

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MATH 2131 3.00 MS2 Assignment 3 Total marks = 50 Question 1: Let ( X,Y ) have the joint pdf f X,Y ( x,y ) = ( e - y , 0 < x < y < , 0 , otherwise. (a) (4 marks). Find E ( XY ). (b) (6 marks). Find Cov( X,Y ). (c) (4 marks). Find ρ X,Y . Question 2: Let X 1 ,...,X n be a random sample on r.v. X whose mean is μ and variance is σ 2 . Let ¯ X = n i =1 X i /n and S 2 = n i =1 ( X i - ¯ X ) 2 / ( n - 1). Show that (a) (2 marks). E ( ¯ X ) = μ . (b) (2 marks). Var( ¯ X ) = σ 2 /n . (c) (5 marks). E ( S 2 ) = σ 2 . Question 3: (3 marks). The number of transactions handled by a bank teller in a day is a r.v. X with a mean of 80 and a standard deviation of 5. Using Chebyshev’s inequality, what can be said about the probability that the teller will handle between 70 and 90 transactions in a day? Question 4: (4 marks). Assume that the number of games, X , that a baseball relief pitcher will save in n games follows a binomial distribution, i.e., X Binomial( n,p
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Unformatted text preview: ). In a five-game playo± series between equally matched teams, the number of games played is a r.v., N , with pmf p N ( n ) = ( ± n-1 2 )± 1 2 ) n-1 , n = 3 , 4 , 5 , otherwise. Find the expected number of saves this relief pitcher will make in this series. 1 Question 5: (3 marks). If X and Y are r.v.’s with equal variances, find Cov( X + Y,X-Y ). Question 6: (6 marks). Let X , Y , and Z be uncorrelated r.v.’s with variances σ 2 X , σ 2 Y , and σ 2 Z , respectively. Let U = Z + X and V = Z + Y . Find ρ U,V . Question 7: Let (X,Y) have the joint pdf f X,Y ( x,y ) = ( 8 xy, < x < y < 1 , , otherwise. (a) (7 marks). Find E ( X | Y = y ) and Var( X | Y = y ). (b) (4 marks). Find the pdf of r.v. E ( X | Y ). 2...
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This note was uploaded on 02/29/2012 for the course MATH 2131 taught by Professor Peskun during the Spring '11 term at York University.

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a3su112131 (1) - In a five-game playo± series between...

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