MA519_AUG04

# MA519_AUG04 - X and Y . (c) Find ONE of P ( X = 20) , P ( Y...

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QUALIFYING EXAMINATION AUGUST 2004 MATH 519 - Profs. Davis and Viens Do not do arithmetic. I ( A ) is the random variable which equals one on the set A and zero on the complement of A . Each problem is worth twenty points. 1. Customers entering a store spend nothing half the time and the other half the time they spend a continuous uniform on the interval (0 , 1) number of dollars. Use Chebyshev’s inequality to bound the probability that the average number of dollars spent by the ﬁrst one hundred customers to enter the store is between . 23 and . 27. 2. A die is rolled repeatedly. Let A k be the event that the sum of the k th and the ( k +1) st rolls is two. Let B k be the event that the sum of the k th and the ( k +1) st rolls is seven. Let X = 100 X k =1 I ( A k )andlet Y = 100 X k =1 I ( B k ). (a) Are the events A k , k =1 , 2 , ··· independent? Are the events B k , k =1 , 2 , ··· independent? Brieﬂy justify these answers. (b) Find the mean and variance of
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Unformatted text preview: X and Y . (c) Find ONE of P ( X = 20) , P ( Y = 20). 3. Suppose that A and B are two points chosen at random in the unit circle in R 2 . Find the probability that the circle centered at A with radius AB (i.e., radius equals the distance between A and B ) is completely contained within the unit disc. 4. Two points are picked at random from the unit square { ( x,y ) : 0 ≤ x ≤ 1 , ≤ y ≤ 1 } . Find the probability that the (inﬁnite) straight line through the two points intersects the line segment { ( x,y ) : x = 1 , ≤ y ≤ 1 } . 5. Ten white and ten black balls are divided into two boxes. What is the division that maximizes the probability that if one of the boxes is picked at random and then one ball is drawn from the box at random, the chosen ball is white? Justify your answer. 1...
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## This document was uploaded on 01/25/2012.

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