# MA519_JAN03 - QUALIFYING EXAMINATION JANUARY 2003 MATH 519...

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QUALIFYING EXAMINATION JANUARY 2003 MATH 519 - Prof. Sellke Grading: Ten points for each problem 1. Find the greatest possible value of E ( XY ), if X is Exponential ( λ =1 )and Y is discrete uniform on { 1 , 2 } . Justify your answer. 2. Let X 1 ,X 2 ,... be iid Exponential ( λ = 1) random variables. Let M n =max { X 1 ,...,X n } . Find constants c n so that the diﬀerence D n = M n - c n converges in distribution, and ﬁnd the limiting distribution. 3. Let X 1 ,X 2 ,... be iid Exponential ( λ = 1) random variables, and let L n = min { X 1 ,...,X n } . Show that the quotient L 2 n L n converges in distribution, and ﬁnd the c.d.f. of the limiting distribution. 4. Let ( X,Y ) be a random point in R 2 , distributed uniformly on the interior of the unit circle. Find the density of T = X + Y . 5. Each day, starting on January 1, 2002, Professor Sellke has recorded the value of a standard normal random variable, generated according to the precepts of Professor
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## This document was uploaded on 01/25/2012.

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