Let say that $X_1,\dots ,X_m$ are independent random variables following Poisson law of parameter $λ_1,\dots, λ_m$.
I want to calculate the probability $P(X_i=j|X_1+X_2=k)$
\begin{align*} P(X_1=j|X_1+X_2=k)=\frac{P(X_1=j;X_1+X_2=k)}{P(X_1+X_2=k)} \end{align*}
As far as $X_1$ and $X_1 + X_2$ are not independent but $X_1$ and $X_2$ are, can we write:
\begin{align*} P(X_i=j|X_1+X_2=k)=\frac{P(X_1=j)P(X_2=k-j)}{P(X_1+X_2=k)}? \end{align*}
Then it would end as:
\begin{align*} P(X_i=j|X_1+X_2=k)=\frac{\lambda_1^j\lambda_2^{k-j}}{(\lambda_1+\lambda_2)^{k-j}}? \end{align*}