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## Main Question or Discussion Point

I'm reading a stat textbook and it says the following:

Let a discrete-time random walk be defined by X

(i) E(X

(ii) Var(X

However, the textbook doesn't have a lot of justifications for these results and I don't understand why (i) and (ii) are necessarily true here.

For example, E(X

Can someone please explain in more detail?

Thanks a lot!

Let a discrete-time random walk be defined by X

_{t}= X_{t-1}+ e_{t}, where the e_{t}'s are i.i.d. normal(0,σ^{2}). Then for t≧1,(i) E(X

_{t}) = 0(ii) Var(X

_{t}) = t σ^{2}However, the textbook doesn't have a lot of justifications for these results and I don't understand why (i) and (ii) are necessarily true here.

For example, E(X

_{t}) = E(X_{t-1}+e_{t}) = E(X_{t-1}) + E(e_{t}), but how can you calculate E(X_{t-1})?Can someone please explain in more detail?

Thanks a lot!