A walker starts at 0 and takes 4 steps, each independently +1 or −1 with probability 1/2. Let M be the highest position reached, counting the start. What is E[M]?

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A walker starts at $0$ and takes $4$ steps, each independently $+ 1$ or $- 1$ with probability $\frac{1}{2}$ . Let $M$ be the highest position reached, counting the start. What is $E [M]$ ?

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Hint

Use the reflection principle to find P(M ≥ k).

Solution

The final position $S$ takes values $- 4, - 2, 0, 2, 4$ with probabilities $\frac{1}{16}, \frac{4}{16}, \frac{6}{16}, \frac{4}{16}, \frac{1}{16}$ .
By reflection, $P (M \geq k) = P (S \geq k) + P (S > k)$ , which gives $\frac{10}{16}$ , $\frac{6}{16}$ , $\frac{2}{16}$ and $\frac{1}{16}$ for $k = 1, 2, 3, 4$ .
Then $E [M] = \sum P (M \geq k)$ is $\frac{10 + 6 + 2 + 1}{16}$ .
That equals $\frac{19}{16}$ .

Answer: $\frac{19}{16}$

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