09mid - Statistics 150(Stochastic Processes Midterm Exam...

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Statistics 150 (Stochastic Processes): Midterm Exam, Spring 2009. J. Pitman, U.C. Berkeley. 1. A sequence of random variables X 1 , X 2 , . . . , each with two possible values 0 and 1, is such that P ( X 1 = 1) = p 1 and for each n 1 P ( X n +1 = 1 | X 1 , . . . , X n ) = (1 - θ n ) p 1 + θ n S n /n where ( θ n ) is a sequence of parameters with 0 θ n 1, and S n := X 1 + · · · + X n . Find and prove a formula for P ( X n = 1) in terms of p 1 and θ 1 , θ 2 , . . . . 2. Consider a Markov chain ( X n ) with transition matrix P . For 1 m < n find and explain a formula for the conditional distribution of X m given X 0 = i and X n = k in terms of the entries of appropriate matrix powers of P . 3. Consider a p , 1 - p walk ( S n ) started at S 0 = a and run until the time T when it first hits either 0 or b for some positive integers 0 a b . Assuming p 6 = 1 / 2, justify an application of Wald’s identity to derive a simple formula for E ( T | S 0 = a ) in terms of p and the known solution of the gambler’s ruin problem for an unfair coin, denote it h ( p, a, b ) := P ( S T = b | S 0 = a ) .
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