final-prac-1 - Statistics 426: Final Exam Moulinath...

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Unformatted text preview: Statistics 426: Final Exam Moulinath Banerjee April 28, 2004 Announcement: The final exam carries 80 points. Half of what you score contributes to your grade in the course. (1) . (a) Let Z 1 ,Z 2 ,Z 3 be i.i.d. N(0,1) random variables. Let R = q Z 2 1 + Z 2 2 + Z 2 3 . Find the density function of R . (Hint: Can you write down the density function of R 2 ?) (b) Suppose that X ( , ) and Y ( , ) where ,, > 0. Let U = X + Y and V = X X + Y . (i) Show that the joint density of ( X,Y ) is f ( x,y ) = + ( 1 )( 2 ) e- ( x + y ) x - 1 y - 1 , x > ,y > . (ii) Compute the joint density of U and V . Deduce that they are independent and write down their marginal densities. (c) Let X be a random variable with distribution function F ( x ). Let f ( x ) be the density function of X . Evaluate R - F ( x ) f ( x ) dx . (Hint: How is F ( X ) distributed?) ( 5 + 10 + 5 = 20 points) (2) . (a) Consider two groups of patients, say Group A and Group B. The number of patients(2) ....
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This note was uploaded on 04/14/2010 for the course STATS 426 taught by Professor Moulib during the Winter '08 term at University of Michigan.

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final-prac-1 - Statistics 426: Final Exam Moulinath...

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