if i have 2 intervals on the same PP
[15,25] [16,36]
the 2 pmf are still independent ?
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[15,25] [16,36]
the 2 pmf are still independent ?
$$ (X,Y) = (U+V,V+W) $$ where $V$ is the count in the overlap, and $U,V,W$ are independent and Poisson-distributed.
\begin{align} & \Pr(X=x\ \&\ Y=y) \\[8pt] = {} & \sum_{z\,=\,0}^{\min\{\,x,\,y\,\}} \Pr(V=z)\Pr(U = x-z)\Pr(W=y-z) \\[8pt] = {} & \sum_{z\,=\,0}^{\min\{\,x,\,y\,\}} \frac{\beta^z e^{-\beta}}{z!} \cdot\frac{\alpha^{x-z} e^{-\alpha}}{(x-z)!} \cdot \frac{\gamma^{y-z} e^{-\gamma}}{(y-z)!} \end{align}
In case $\gamma=0,$ this has a closed form: \begin{align} & \Pr(X=x\ \&\ Y=y) \\[8pt] = {} & \Pr(V=y\ \&\ U=x-y) = \frac{\gamma^y e^{-\gamma}}{y!} \cdot \frac{\alpha^{x-y} e^{-\alpha}}{(x-y)!}. \end{align}
(And if $\beta=0$ then $X,Y$ are independent and the sum has only the one term in which $z=0.$)