$P$ is a Poisson Process with rate $\lambda$. Let $T_1$ be the time of the first event and let $T_2$ be the time of the from the first to the second event. Let $Y = \frac{T_1}{T_1+T_2}$. Find the density of $Y$.
I think I should find the CDF first and then take the derivative, but I do not know how to find $P(Y\leq t)$. Do I need to find the joint density?
Best Answer
\begin{align} & \Pr(Y\le t) = \Pr( T_1 \le t(T_1+T_2)) = \Pr((1-t)T_1\le tT_2) \\[8pt] = {} & \iint\limits_{(u,v)\,:\,(1-t)u\,\le\,tv} e^{-u} e^{-v} \, d(u,v) \\[8pt] = {} & \int_0^\infty \left( \int_0^{tv/(1-t)} e^{-u} e^{-v} \, du \right) \, dv \\[8pt] = {} & \int_0^\infty \big( 1 - e^{-tv/(1-t)} \big) e^{-v} \, dv \\[8pt] = {} & 1 - (1-t) = t \quad \text{provided $0<t<1$}. \\ & \text{So the distribution is uniform on $[0,1]$.} \end{align}