This is probably easy, but right now I can't figure it out, so bear with me. The question is:

Let $\{N(t),t\ge 0\}$ be a homogeneous Poisson process on $(0,\infty)$ with rate $\lambda$. Let $\{S_i, i=1,2,...\}$ be the points of the Poisson process, such that $S_1 <S_2 < S_3 <...$. Define $S_0=0$. The distribution of $S_1$ is Exp($\lambda$). For $x\in[0,T]$ compute $P[T-S_n>x|N(T)=n]$, $E[T-S_n|N(t)=n]$ and $E[T-S_{N(T)}]$.

There is a solution-manual for this, and I've tried to read it, but I don't understand everything in it. And that is why I need some help.

For $P[T-S_n>x|N(T)=n]$ it says that "by using the order statistic-property the answer is $(\frac{T-x}{T})^n$. I understand that we want to compute the probability that all the $n$ points falls into the interval $(0,T-x)$, but from there to how we get $(\frac{T-x}{T})^n$ I don't understand.

Also, the solution-manual says that when you compute $E[T-S_n|N(t)=n]$, you should use the "standard identity" of $E(X)=\int_{0}^{\infty} P(X>x) dx$. What do they mean? I thought that the standard identity was $E(X)=\int_{0}^{\infty} x f(x) dx$?

(If you want to vote down my question, fine, but if the reason is that the question is easy for you, please answer it before you do that)


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