Suppose two Poisson processes. For example, during the time interval, $\Delta t_{1} = t_{1} - t_{o} = 50\mu s$ , $x$ photons are incident on a detector with rate $\lambda_{1} = 10$x$10^4 s^{-1}$. At time point, $t_{1}$, a second process begins in which, during the time interval, $\Delta t_{2} = t_{2} - t_{1} = 50\mu s$ , $y$ photons are incident on the same detector with rate $\lambda_{2} = 6$x$10^4 s^{-1}$.
Let $X$ and $Y$ be two independent Poisson random variables described by $X$ ~ Pois($\lambda_{1}\Delta t_1$) and $Y$ ~ Pois($\lambda_{2}\Delta t_2$). And let $Z$ be a ratio distribution defined as $Z = X/(X+Y)$.
[1] What is the general distribution of $Z$ for $X+Y>0$? its standard deviation? and how are both derived?
Next, suppose we know the total number of photons, $n=x+y$ , over the time interval $\Delta t = \Delta t_{1} + \Delta t_{2} = 100\mu s$ ; e.g., $n=10$.
We would like to predict the probability distribution for observing an $(x,y)$ pair given $n$ and the knowledge that both $x$ and $y$ were drawn from Poisson distributions with rates $\lambda_1 \Delta t_1$ and $\lambda_2 \Delta t_2$, respectively.
[2] What is the new distribution for $Z|n$? its standard deviation? and how are both derived?