Conditional expectation with a Poisson Process?

Question

There is a Poisson process described by the constant intensity $\lambda >0$.

Given that at time $T$ we have encountered the tenth arrival, find the conditional expectation $E[W_1 W_2 \dots W_9\mid W_{10}=T]$.

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I know that $$\frac{W_r}{t}|W_n = T$$ is described by the beta distribution with parameters $\alpha = r, \beta=n-r$ and I therefore know how to calculate each individual expectation. But I'm not sure how to deal with this concerning the product of the expectations because I don't quite understand how the independence of the events.

Thanks for any help.

Answer 1

You have that the set $\{W_{1},\ldots W_{n}\}$ has the same distribution that a vector of a independent random variable $Y_{1}, \ldots, Y_{n}$ with a uniform distribution in the interval $[0,t]$.

So , conditioning on $W_{10}=T$ the set $\{W_{1},\ldots W_{n}\}$ is distributed like: $TY_{1}, \ldots, TY_{n}$ iid Uniform on $(0,1)$, then $E(TY_{n})=\dfrac{T}{2}$.

Therefore,

$$ E(W_{1},\ldots, W_{9}|W_{10}=T)=E(TY_{1}, \ldots, TY_{9})=(\dfrac{T}{2})^{9}$$.

I hope, that like my answer.