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Suppose that $S_n$ is a simple random walk started at $0$, so that $S_n = X_1 + \dots + X_n$ where $X_j$'s are iid random variables taking values $1$ and $-1$ with probability $p = 3/4$ and $q=1/4$ respectively. Compute $\mathbb{E}(S_{n+2} | S_n = 2)$.

My immediate approach was to do: $$ \mathbb{E}(S_{n+2} | S_n = 2) = \mathbb{E}(X_{n+2} | S_n = 2) + \mathbb{E}(X_{n+1} | S_n = 2) + \mathbb{E}(S_{n} | S_n = 2) = \mathbb{E}(X_{n+2}) + \mathbb{E}(X_{n+1}) + 2 = 1/2 +1/2 +2 = 3 $$ But the solutions state: $$ \mathbb{E}(S_{n+2} | S_n = 2) = (2+2) \times 9/16 + 2 \times 3/16 -2 \times 1/16 = 2.5 $$ without further explanation. I have no idea how to come up with this result, can someone point out my mistake and explain how to get the correct result? Thank you.

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  • $\begingroup$ I think your answer is correct. $\endgroup$ – Kavi Rama Murthy Apr 18 at 7:29

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