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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?

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    It's not clear if the fact that the two times intervals are consecutive is of any significance. Actually, the whole first paragraph seems pretty redundant, is it? And, rather, in the second paragraph, $X \approx $ Pois ($\lambda_1)$ ... here $\lambda_1$ corresponds to $\lambda_1 \Delta t_1$ in the first paragraph...2012-06-04
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    Conditionally on $X+Y=n$, the distribution of $X$ is binomial $(n,p)$, with $p=\lambda_1\Delta t_1/(\lambda_1\Delta t_1+\lambda_2\Delta t_2)$.2012-06-11
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    @did Having difficulty visualizing the solution. How is the problem equivalent to a coin tossing experiment? If knowing $n$ simply reduces the problem to a binomial, $Z \sim X|n / (X|n + n - X|n) = X|n / n$ . Hence, $\sigma_Z = \sqrt{p(1-p)/n}$ . Does order not matter (e.g., $x$ photons before $y$)?2012-06-11
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    See my answer below.2012-06-11
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    Cross-posted on [stats](http://stats.stackexchange.com/questions/30226).2012-06-11

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