Question

Problem 5.1 (10 points) Let (Sn)n?0 be a simple random walk starting in 0 (i.e. S0 = 0) with p = 0.4 and q = 1 ? p = 0.6. Compute explicitly the following probabilities: (i) P0[S2 = ?2, S5 = ?3], (ii) P0[S2 = ?2, S4 = 0, S5 = 2], (iii) P0[T5 ? 7], where Tk := min{n > 0 : Sn = k} for all k ? Z. Problem 5.2 (10 points) Let (Sn)n=0,1,2,... be a simple, symmetric random walk with starting value S0 = s ? R. (a) Show that E[Sn] = s for all n ? 0. (b) Show that E[Sn+1 | Sn] = Sn for all n ? 0. (c) Suppose that (Sn)n=0,1,2,... denotes the profit and loss from $1 bets of a gambler with initial capital S0 = s who is repeatedly playing a fair game with 50% chances to win or lose her stake. What are the interpretations of the results in (a) and (b)?

          Problem 5.1 (10 points) Let (Sn)n?0 be a simple random walk starting in 0 (i.e. S0 = 0) with p = 0.4 and q = 1 ? p = 0.6.
Compute explicitly the following probabilities:
(i) P0[S2 = ?2, S5 = ?3],
(ii) P0[S2 = ?2, S4 = 0, S5 = 2],
(iii) P0[T5 ? 7], where Tk := min{n > 0 : Sn = k} for all k ? Z.
Problem 5.2 (10 points) Let (Sn)n=0,1,2,... be a simple, symmetric random walk with starting value S0 = s ? R.
(a) Show that
E[Sn] = s for all n ? 0.
(b) Show that
E[Sn+1 | Sn] = Sn for all n ? 0.
(c) Suppose that (Sn)n=0,1,2,... denotes the profit and loss from $1 bets of a gambler with initial capital S0 = s who is repeatedly playing a fair game with 50% chances to win or lose her stake. What are the interpretations of the results in (a) and (b)?

Problem 5.1 (10 points) Let (Sn)n?0 be a simple random walk starting in 0 (i.e. S0 = 0) with p = 0.4 and q = 1 ? p = 0.6.
Compute explicitly the following probabilities:
(i) P0[S2 = ?2, S5 = ?3],
(ii) P0[S2 = ?2, S4 = 0, S5 = 2],
(iii) P0[T5 ? 7], where Tk := minn > 0 : Sn = k for all k ? Z.
Problem 5.2 (10 points) Let (Sn)n=0,1,2,... be a simple, symmetric random walk with starting value S0 = s ? R.
(a) Show that
E[Sn] = s for all n ? 0.
(b) Show that
E[Sn+1 | Sn] = Sn for all n ? 0.
(c) Suppose that (Sn)n=0,1,2,... denotes the profit and loss from 1 bets of a gambler with initial capital S0 = s who is repeatedly playing a fair game with 50% chances to win or lose her stake. What are the interpretations of the results in (a) and (b)?

Added by David S.

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