Understanding the Variance of a One-Dimensional Random Walk

[grad]

Hi,

I know that the expectation E(S_n) 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(S_n²) = n.

Why is that? Can anyone prove it?

Thank you very much!

 

[Pere Callahan]

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

 

[grad]

Var(S_n) = E(S_n²) = E(Z₁² + Z₂² + Z₃² + ... + Z_n²) =^* E(Z₁²) + E(Z₂²) + ... + E(Z_n²) = 1 + 1 + ... + 1 _{(n times)} = n

*variables are independent and uncorrelated

Is this correct then?

 

[Pere Callahan]

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 $$\mathbb{E}\left[S_n^2\right]=n$$.

 

[grad]

Thank you!

 

[Pere Callahan]

You're welcome