I have a binomial $CNT\sim\mathcal{B}(n,p)$, and a poisson $P\sim \mathcal{P}(\lambda)$.
Now I'm looking at $CNT$ conditional on $P$:
What’s the expected value, $E(B|P)$? What is the variance $Var(B|P)$? What is the covariance $Cov(B,P)$? Are there closed forms, or must we always go through the sums?
After feedback in the comments, here's what I have. Going through the sums:
$$ E(\textbf{CNT}|A^*) = \sum_{w=0}^{w^*} \sum_{v=-w}^{+w} vP(\text{CNT}=v|A^*=w) P(A^*=w) $$ $$ = \sum_{w=0}^{w^*} \sum_{k=0}^{w} (2k-w) P(\text{CNT}=(2k-w)|A^*=w) P(A^*=w) $$
$$ =\sum_{w=0}^{w^*} \sum_{k=0}^{w} (2k-w) \binom{w}{k} p^{k} (1-p)^{w-k} P(A^*=w) $$
where $w^* = 4\lambda$, $\text{CNT} \sim{B}(w,p)$, and $A^*\sim{P}(\lambda)$.
This, and the corresponding for variance, yields correct numerical values. What I wonder is whether or not there are closed forms for $E(CNT|A^*)$, $Var(CNT|A^*)$, and $Cov(CNT,A^*)$.