Evaluating $\mathbb E[K\mid N]$ for $N \sim \operatorname{Poisson}(\lambda)$ and $f_{K\mid N}(k\mid n)=\binom nkp^k(1-p)^{n-k}$

Question

Consider the random variables $N \sim \operatorname{Poisson}(\lambda)$ and $K$ whose distribution we don't know. (As far as I know we can't assume that they are independent)

Furthermore let

$$f_{K\mid N}(k\mid n)=\binom nkp^k(1-p)^{n-k}$$

How can I determine $\mathbb E[K\mid N]$?

I calculated $\mathbb E[K\mid N]=\sum_{k=0}^nk\binom nk p^k (1-p)^{n-k}$ but I think that this is $\mathbb E[K\mid N=n]$ and not $\mathbb E[K\mid N]$.

What about $\mathbb E[K]?$

Answer 1

You have $\operatorname E(K\mid N=n) = \sum_{k=0}^n k \binom n k p^k(1-p)^{n-k} = np.$

You can conclude that $\operatorname E(K\mid N) = Np.$

And $\underbrace{\operatorname E(K) = \operatorname E(\operatorname E(K\mid N))}_\text{law of total expectation} = \operatorname E(Np) = p \operatorname E(N) = p\lambda.$