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MA519_JAN01 - MA 519 Introduction to Probability January...

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MA 519 Introduction to Probability January 2000, Qualifying Examination Instructor: Yip This qualifying exam contains five questions. By all means simplify your answers as much as possible. It might be useful to know that for any positive integers m and n , Z 1 0 x m (1 - x ) n dx = m ! n ! ( m + n + 1)! A normal table is provided at the end. 1. There are n people among whom are A and B . They stand in a row randomly. What is the probability that there will be exactly r people between A and B ? What is the corresponding probability if they stand in a circular ring? (In this case, consider only the arc going from A to B in the positive (i.e. counter-clockwise) direc- tion.) 2. Consider a large collection of coins such that the probability p of a coin giving a head is itself a random variable which is uniformly distributed in [0 , 1]. Let X be the total number of heads in n tossing of the coins. Find P ( X = i ) ( i = 0 , 1 , . . . n ) in the following two situations: (a) Pick a coin at random and then toss this coin n times.
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