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HW 4 - P X n& 1 = i n& 1;X = i j X n = i 3 Let X = f...

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APMA 1200, 2010: Assignment #4 1. Assume that S = f S n g is a simple random walk starting at the origin. That is, S n = n X i =1 Y i ; S 0 = 0 where Y 1 ; Y 2 ; : : : are iid random variables with P ( Y i = 1) = p; P ( Y i = ° 1) = q = 1 ° p: (a) For every 0 ± k such that 2 k ± n , argue that P ( S 1 ² 0 ; S 2 ² 0 ; ³ ³ ³ ; S n ° 1 ² 0 j S n = n ° 2 k ) = n ° 2 k + 1 n ° k + 1 (b) Suppose n = 2 m is an even number. Argue that P ( S 1 6 = 0 ; S 2 6 = 0 ; : : : ; S 2 m ° 1 6 = 0 ; S 2 m = 0) = 2 m ° 2 m ° 2 m ° 1 ± p m q m : Hint: For part (a), imagine the random walk as if it started from ( ° 1 ; ° 1) , and mimic the example of re°ection principle that was given in class. For part (b), use part (a) to compute the NUMBER of paths that satis±es S 1 6 = 0 ; : : : ; S n ° 1 6 = 0 ; S n = 0 . And note that each of these path has probability p m q m . 2. Argue that for any Markov chain X = f X n : n ² 0 g , P ( X n + k = j k ; : : : ; X n +1 = j 1 ; X n ° 1 = i n ° 1 ; : : : ; X 0 = i 0 j X n = i ) = P ( X n + k = j k ; : : : ; X
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Unformatted text preview: P ( X n & 1 = i n & 1 ;: :: ;X = i j X n = i ) 3. Let X = f X n g be a Markov chain with state space S and transition probability matrix P = [ P ij ] . Let Y be a Markov chain with state space W and transition probability matrix Q = [ Q kl ] . Furthermore, assume that X and Y are independent. Then the two-dimensional process Z = f Z n g with Z n : = ( X n ;Y n ) is also a Markov chain (you do not need to prove this, but at least try to understand it intuitively). (a) Identify the state space of Z . (b) Identify the transition probability matrix of Z . 1...
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