Finding the joint unconditional distribution of $X$ and $Y=N-X$ for $X\sim \mathrm{Bin}(N,p)$ and $N\sim \mathrm{Pois}(\lambda)$. The question asks to find the unconditional joint distribution of $X$ and $Y=N-X$, given that 


*

*$N$ has a Poisson distribution with parameter $\lambda$, and

*$X$ has a conditional distribution $\mathrm{Bin}(N,p)$.


I have worked out that $X$ has an unconditional distribution $\mathrm{Pois}(p\lambda)$ though I'm not sure that is of much help.
I am having trouble beginning. The steps I used to find the single unconditional distribution were straightforward, but I'm not sure how to go about it for the joint distribution. 
Thanks in advance. 

You have been given the conditional probability mass of $X\mid N$, so why not use it?
$\begin{align}
\mathsf P(X=k, Y=h) & = \mathsf P(X=k, N-X=h)
\\ & = \mathsf P(X=k, N=h+k)
\\ & = \mathsf P(X=k\mid N=h+k)\cdot \mathsf P(N=h+k)
\\ & = \binom{h+k}{k}p^{k}(1-p)^{h}\cdot\frac{\lambda^{h+k}e^{-\lambda}}{(h+k)!}
\end{align}$
