Variance in Random Walk

[grad]

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 the Wikipedia (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!

[Pere Callahan]

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

[grad]

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?

[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:smile: