# Conditional expectation for a simple random walk

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.

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