A symmetric random walk starts at 0. Each step is independently +1 or -1 with equal probability. What is the expected position after 100 steps?

Apply linearity. Position after 100 steps is S100 = i=1100 Xi with E[Xi]=0.

Symmetric Walk Mean

Expected Value

Random Walk

Easy

A symmetric random walk starts at

0

. Each step is independently

+ 1

- 1

with equal probability. What is the expected position after

100

steps?

Hint 1

Each step has expected value $0$ by symmetry.

Hint 2

Use linearity of expectation over all 100 steps.

Show worked solution

1. Apply linearity
Position after $100$ steps is $S_{100} = \sum_{i = 1}^{100} X_{i}$ with $E [X_{i}] = 0$ .
$E [S_{100}] = i = 1 \sum 100 E [X_{i}] = 0$

Answer: 0

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