## Problem about the sum of independent exponential variable

I repost the problem because I think the answers did not fully solve the problem.

Let $X_1,\ldots,X_{n}$ be independent exponential variables with mean 1, and let $S_k = X_1+\cdots+ X_k$, it is not hard to get $\mathbb{E}(S_k)=k$.

Let random variable $Y_k=|S_k-k|$,

My first question is: what is the probability of $Y>t$ for some $t>0$, in another word: $\Pr(Y>t)$?

Define another random variable $Z=\max_{k=1}^n Y_k$

The question is: how to calculate $\mathbb {E} (Z)$.

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As Dider Piau gave when you asked at math.stackexchange, there is a natural asymptotic for $E(Z)$ when $k$ is large from the Brownian motion approximation. I wouldn't be hopeful that there is a nice exact expression for $E(Z)$ for general $k$. Do you really need an exact answer for fixed $k$? If not, explain a bit more what sort of answer would be useful for your purposes. – James Martin Jan 27 2012 at 16:23