MA519_JAN99 - E ( U | V = v ). (20) 4. Let Z 1 ,Z 2 , ,Z n...

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QUALIFYING EXAMINATION JANUARY 1999 MATH 519 - Prof. DasGupta (20) 1. A group of n people are lined up in a row at random. Let S denote a randomly selected nonempty subset of these n people, selected from the 2 n - 1 possible nonempty subsets of the full set of n people. Find the probability that the members of S occupy consecutive positions in the line-up. (20) 2. Suppose X 1 ,X 2 , ··· ,X k are k IID Poisson random variables with mean 1, and n 1 ,n 2 , ··· ,n k are k nonnegative integers. Characterize all k -tuples ( n 1 ,n 2 , ··· ,n k ) such that k X i =1 n i X i has also a Poisson distribution. (20) 3. Take two IID Uniform [0 , 1] random variables X , Y . Let U = X + Y, V = XY . Find an expression for
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Unformatted text preview: E ( U | V = v ). (20) 4. Let Z 1 ,Z 2 , ,Z n be n IID N (0 , 1) variables. We now dene X i = Z i if | Z i | > 1 = 0 if | Z i | 1 . Consider now the random variable T n T n = n X i =1 X i . Approximately calculate P ( T n > 10) when n = 50. (20) 5. A couple have agreed on having children until they have at least 2 boys and at least 2 girls. Assume that childbirths are mutually independent and that at each birth, a male or a female child are equally likely. What is the most likely number of children this couple will have?...
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