Arrivals are described by a Poisson process with a constant intensity $\lambda.$
We are asked to describe the conditional density of the ratio $\frac{W_9}{W_{10}}$, given that at the time $T$ the process has encountered its tenth arrival.
I know that the ratio $\frac{W_k}{W_l}$ is described by a Beta distribution with parameters $\alpha=k$ and $\beta=l-k$. I just need help understanding what the conditional density of the ratio is. Is it strictly Beta$(9,1)$?
Thanks for your help!
Conditional on $N(T)=10$, the joint distribution of $(W_1,\ldots,W_{10})$ is the same as that of $(U_{(1)},\ldots,U_{(10)})$, where $U_1,\ldots,U_{10}\sim\mathsf U(0,T)$, which has joint density $$f(s_1,\ldots,s_{10})=\frac{10!}{T^{10}}, 0<s_1<\cdots<s_n<T. $$ The joint distribution of $(U_{(9)},U_{(10)})$ is then $$f_{U_{(9)},U_{(10)}}(u,v) = \frac{10!}{T^{10}}\left(\frac{u^8}{8!}\right), 0<u<v<T, $$ so we may compute the distribution of $W:=\frac{U_{(9)}}{U_{(10)}}$ by $$f_W(w) = \int_{\mathbb R} |v| f_{U_{(9)},U_{(10)}}(wv,v)\ \mathsf dv =\int_0^T v\frac{10!}{T^{10}}\left(\frac{(wv)^8}{8!}\right)\ \mathsf dv = 9w^8, 0<w<1. $$