# Variance in Random Walk

## Main Question or Discussion Point

Hi,

I know that the expectation E(Sn) for a one-dimensional simple random walk is zero. But what about the variance?

I read in http://en.wikipedia.org/wiki/Random_walk#One-dimensional_random_walk" that the variance should be E(Sn2) = n.

Why is that? Can anyone prove it?

Thank you very much!

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Just write down the definition of $S_n$ and you will be able to answer your question yourself.

Var(Sn) = E(Sn2) = E(Z12 + Z22 + Z32 + ... + Zn2) =* E(Z12) + E(Z22) + ... + E(Zn2) = 1 + 1 + ... + 1 (n times) = n

*variables are independent and uncorrelated

Is this correct then?

This is almost correct. $S_n$ is defined to be $Z_1+\ldots +Z_n$, where the $Z_i$ are independent (or at least uncorrelated) with mean zero and variance one. It follows that
$$S_n^2 = \sum_{i,j=1}^n{Z_i Z_j}$$
and not, as you wrote,
$$S_n^2 = \sum_{i=1}^n{Z_i^2}$$

However, using independence of the $Z_i$ you can still do a similar computation to prove $$\mathbb{E}\left[S_n^2\right]=n$$.

Thank you!

You're welcome 