Unformatted text preview: ion probability p(i, i + 1) = (N i)/N , and p(i, i 1) = i/N for 0 i N . 72 CHAPTER 1. MARKOV CHAINS Let µn = Ex Xn . (a) Show that µn+1 = 1 + (1 2/N )µn . (b) Use this and
induction to conclude that
From this we see that the mean µn converges exponentially rapidly to the
equilibrium value of N/2 with the error at time n being (1 2/N )n (x N/2).
1.54. Prove that if pij > 0 for all i and j then a necessary and su cient
condition for the existence of a reversible stationary distribution is
pij pjk pki = pik pkj pji for all i, j, k Hint: ﬁx i and take ⇡j = cpij /pji .
Exit distributions and times
1.55. The Markov chain associated with a manufacturing process may be described as follows: A part to be manufactured will begin the process by entering
step 1. After step 1, 20% of the parts must be reworked, i.e., returned to step
1, 10% of the parts are thrown away, and 70% proceed to step 2. After step 2,
5% of the parts must be returned to the step 1, 10% to step 2, 5% are scrapped,
and 80% emerge to be sold for a proﬁt. (a) Formula...
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