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final-prac-1

# final-prac-1 - Statistics 426 Final Exam Moulinath Banerjee...

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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 Γ( α, λ ) and Y Γ( β, λ ) where α, β, λ > 0. Let U = X + Y and V = X X + Y . (i) Show that the joint density of ( X, Y ) is f ( x, y ) = λ α + β Γ( α 1 ) Γ( α 2 ) e - λ ( x + y ) x α - 1 y β - 1 , x > 0 , y > 0 . (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 in each of these groups is 50. Group A patients were on a blood pressure drug for 4 weeks while Group B patients were on placebo.

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

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