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MA519_AUG09

# MA519_AUG09 - QUALIFYING EXAMINATION August 2009 MA 519(M D...

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QUALIFYING EXAMINATION August 2009 MA 519 (M. D. Ward) 1. Consider n 2 random points, placed uniformly and independently, onto the edge of a circle with circumference 1. [An “arc” denotes a path on the circle.] 1a. (5 pts) Find the density of the length of the arc that connects the ﬁrst point to the closest of the other points. 1b. (3 pts) Find the density of the straight line distance (i.e., not on the circle) between the two points described in part 1a . 1c. (5 pts) Consider the length of the largest arc that contains none of the n points in its interior. Prove that the length of this largest arc goes to 0 in probability as n → ∞ . 2. (5 pts) Consider independent random variables X and Y , with X uniform on (0 , 2) and with Y uniform on (0 , 3). Let M = max( X,Y ) and m = min( X,Y ). Find P ( m 2 > M ). 3. (5 pts) Ten students organize a tournament. Each student competes against each other student exactly once; all competitions are independent. In each competition, both students

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MA519_AUG09 - QUALIFYING EXAMINATION August 2009 MA 519(M D...

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