HW4solns - HW4 Problem Solutions#3.4.1 We are given the state-space S = N(I am using j to denote what the book writes as E j Clearly the

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Unformatted text preview: March 18, 2007 HW4 Problem Solutions #3.4.1. We are given the state-space S = { , . . . , N } . (I am using j to denote what the book writes as E j .) Clearly the transitions are given by: P ij = p I [ j =0] + (1- p ) I [ j = i +1 ≤ N ] if i < N P ij = (1- rp )) I [ j = N ] + rp I [ j =0] if i = N Irreducibility is obvious because 0 7→ 1 7→ ··· 7→ N- 1 7→ N 7→ 0. For aperiodicity of an irreducible chain, it is always sufficient to check that some state (here, N or 0) has positive probability of succeeding itself. The stationary equations are: π j = π j- 1 (1- p ) = π (1- p ) j , j = 1 , . . . , N- 1 π N = π N- 1 (1- p ) + π N (1- rp ) = (1- p ) N rp π and (using the fact that π must be a probability vector) π = p ( π + ··· π N- 1 ) + rp π N = p (1- (1- r ) π N ) = p- (1- p ) N 1- r r π which implies π = p n 1 + (1- p ) N 1- r r o- 1 Then, since items are always inspected in states 0 , . . . , N- 1 and inspected only with probability r in state N , the long-run proportion of items inspected is 1- π N + rπ N = r/ (1 + (1- r )(1- p ) N ). Finally, since we know a proportion p of all ( iid ) items is defective, the long-term or stationary proportion of defective items inspected is just this same overall proportion of inspected items.items inspected is just this same overall proportion of inspected items....
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This note was uploaded on 09/09/2009 for the course STAT 650 taught by Professor Slud during the Spring '09 term at Maryland.

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HW4solns - HW4 Problem Solutions#3.4.1 We are given the state-space S = N(I am using j to denote what the book writes as E j Clearly the

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