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MA519_AUG06 - D 1 D 2 and D 3 be the distances of the darts...

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QUALIFYING EXAMINATION AUGUST 2006 MATH 519 - Prof. Davis Each problem is worth 20 points. 1. A die is rolled until two different numbers appear. Let T be the total number of times the die is rolled. Find ET and also find the probability that exactly one three is rolled up to and including roll number T . 2. Toss a quarter n times. Each time the quarter comes up heads toss a nickel. Let X n equal the number of heads on the quarter, and Y n be the number of heads on the nickel. Find E ( X n + Y n ) and EX n Y n . 3. A circular dartboard has a radius of one foot. Three different darts are thrown at the dartboard. The darts independently hit the board at locations uniformly distributed on the face of the board. Let
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Unformatted text preview: D 1 , D 2 , and D 3 be the distances of the darts to the center of the dartboard (so 0 ≤ D i ≤ 1), and let ≤ D (1) ≤ D (2) ≤ D (3) ≤ 1 be the corresponding order statistics. Find E ( D (2) ) and the joint density of D (1) and D (3) . 4. Ten men and ten women are randomly seated around a round table. Let N be the number of men whose immediate neighbors are both women. Find EN , P ( N = 10), P ( N = 9). 5. Find a function f : R 2 → R 2 such that if U 1 and U 2 are independent uniform (0 , 1) random variables then f ( U 1 ,U 2 ) is uniformly distributed on the triangle { ( x,y ) : 0 ≤ x ≤ 2 , ≤ y ≤ x } ....
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