<s>[INST] sum and difference between two independent Poisson random variables Let $X$ and $Y$ be independent Poisson variables with respective parameters $a$ and $b$. 
What is 


*

*the distribution function of $X+Y$? 

*the conditional distribution of $X$, given $X+Y=n$? 

*same, given $X-Y=n$?

 [/INST] *

*The first problem can be tackled heads on:
$$ \begin{eqnarray}
  \Pr(X+Y=k) &=& \sum_{p=0}^\infty \sum_{q=0}^\infty \mathrm{e}^{-a-b} \frac{a^p}{p!} \frac{b^q}{q!} \mathbf{1}_{p+q=k} = \mathrm{e}^{-a-b}\sum_{p=0}^{k}  \frac{a^p}{p!} \frac{b^{k-p}}{(k-p)!} \\ &=& \mathrm{e}^{-a-b} \frac{1}{k!} \cdot \sum_{p=0}^{k}  \binom{k}{p} a^p  b^{k-p} = \mathrm{e}^{-a-b} \frac{1}{k!} \left(a+b\right)^k
\end{eqnarray}
$$
The last equality follows by the binomial theorem. The expression is the p.m.f. of a Poisson distribution, hence $X+Y \sim \operatorname{Poi}\left(a+ b\right)$.

*The second follows from the first problem. Assuming $0 \leqslant k \leqslant n$:
$$\begin{eqnarray}
    \Pr\left(X=k \mid X+Y=n\right) &=& \frac{\Pr\left(X=k, Y=n-k\right)}{\Pr(X+Y=n)} \\ &=& \frac{\exp(-a) a^k/k! \cdot \exp(-b) b^{n-k}/(n-k)!}{\exp(-a-b) \cdot (a+b)^n/n!} \\
 &=& \binom{n}{k} \left(\frac{a}{a+b}\right)^k \left(1-\frac{a}{a+b}\right)^{n-k}
\end{eqnarray}
$$
The latter is the pmf of a binomial distribution, i.e. $X \mid X+Y=n \sim \operatorname{Bin}\left(n, \frac{a}{a+b} \right)$.

*One has to evaluate $\Pr(X-Y=n)$. The difference of $X-Y$ is known to follow Skellam distribution:
$$
   \begin{eqnarray}
  \Pr(X-Y=n) &=& \sum_{p=0}^\infty \sum_{q=0}^\infty \mathrm{e}^{-a-b} \frac{a^p}{p!} \frac{b^q}{q!} \mathbf{1}_{p-q=n} = \mathrm{e}^{-a-b}\sum_{q=\max(0,-n)}^{\infty}  \frac{a^{n+q}}{(n+q)!} \frac{b^{q}}{q!} \\ &=& \begin{cases} a^n \left(a b\right)^{-n/2} I_n\left(2 \sqrt{a b}\right) & n \geqslant 0 \cr b^{-n} \left(a b\right)^{n/2} I_{-n}\left(2 \sqrt{a b}\right) & n < 0\end{cases} = \left(\frac{a}{b}\right)^{n/2} I_{\vert n\vert} \left(2 \sqrt{a b}\right)
\end{eqnarray}
$$
where $I_n(x)$ is the modified Bessel function of the first kind.
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