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Midterm2

# Midterm2 - X and | X | In what range is each moment...

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ORIE 3500/5500 – Engineering Probability and Statistics II Fall 2010 Midterm 2 The question sheet has two sides. The exam is closed books and notes. You have 90 minutes. All problems have equal weight. Show your work. GOOD LUCK! Problem 1 A continuous random vector ( X, Y ) has a bivariate pdf f X,Y ( x, y ) = 2 3 if x > - 1 , 0 < y < 1 , x y (and equal to zero otherwise). ( a ) Calculate Var( X ) and Var( Y ). ( b ) Calculate the correlation ρ X,Y of X and Y . ( c ) What real number a minimizes the variance of the linear combi- nation aX + Y ? Problem 2 Let X and Y be independent exponential random vari- ables, EX = 5 and EY = 8. ( a ) Define a random variable I by I = 1 if X Y 2 if X > Y . What is E ( I )? ( b ) Calculate the mean and the variance of min( X, Y ). Problem 3 Let X be a continuous random variable with the density f X ( x ) = 1 2 e x if x < 0 e - 2 x if x > 0. ( a ) Compute the moment generating functions of

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Unformatted text preview: X and | X | . In what range is each moment generating function deﬁned? ( b ) Use the moment generating function computed in part ( a ) to calculate the mean and the variance of X . Problem 4 Suppose that X is a nonnegative random variable with EX = 10, Var( X ) = 4. ( a ) Based only on the above information, what is the sharpest upper bound you can obtain on P ( X > 11)? 1 ( b ) Let X 1 ,X 2 ,...,X n be independent and identically disributed random variables each with the same distribution as X , and Y n = ( X 1 + ... + X n ) /n the corresponding sample mean. Based on the above information, how large should the sample size n be to guarantee that P ( Y n > 11) ≤ . 05? 2...
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