For the gamblers ruin model of Section 4.5.1, let Mi denote the mean number of games that

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For the gambler’s ruin model of Section 4.5.1, let Mi denote the mean number of games that must be played until the gambler either goes broke or reaches a fortune of N, given that he starts with i,i = 0, 1,..., N. Show that Mi satisfies M0 = MN = 0; Mi = 1 + pMi+1 + q Mi−1, i = 1,..., N − 1 Solve these equations to obtain Mi = i(N − i), if p = 1 2

= i q − p − N q − p 1 − (q/p)i 1 − (q/p)N , if p = 1 2

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