MA519_JAN94

MA519_JAN94 - QUALIFYING EXAMINATION JANUARY 10, 1994 MATH...

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QUALIFYING EXAMINATION JANUARY 10, 1994 MATH 519 1. (20 points) Let X 1 , X 2 , X 3 and X 4 be independent random variables, each uniformly distributed on the interval ( - 1 , 1). Find (a) P ( X 1 <X 2 <X 3 <X 4 ) (b) P ( X 2 1 < ( X 1 + X 2 ) 2 ) (c) P ( X 2 1 >X 2 2 + X 2 3 ) 2. (20 points) A fair die is rolled repeatedly until it comes up ace. This procedure is repeated 100 times. Find (a) the probability that exactly four threes are rolled in exactly 5 of the 100 repe- titions; (b) the mean and variance of the total number of threes rolled. 3. (20 points) The number of cars arriving at the McDonald’s drive-up window in a given day is a Poisson random variable, N , with parameter λ . The numbers of passengers in these cars are independent random variables, X i , each equally likely to be one, two, three or four. Find the moment generating function of S = N X i =1 X i , the total number of passengers in all the cars. 4. (20 points) Let
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MA519_JAN94 - QUALIFYING EXAMINATION JANUARY 10, 1994 MATH...

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