I'm reading a stat textbook and it says the following:(adsbygoogle = window.adsbygoogle || []).push({});

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!

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# Mean and Variance of Random Walk

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