# Binomial distribution where the number of experiments is binomially distributed

In my setup,

• there are $m$ trials.
• Each trial has a probability $q$ of being selected.
• $N \leq m$ is the number of selected trials
$$\rightarrow N \sim \text{Bin}(q, m)$$

• For each of the $N$ selected trials, the probability of success is $p$

• $K\leq N$ is the number of successful trials
$$\rightarrow (K|N) \sim \text{Bin}(p, N)$$

I have already derived $E[K] = qmp$, and $Var(K)= qmp(1-p) + p^2 m q(1-q)$

However I am stuck in the derivation of $cov(K, N)$. I would appreciate any help to solve this.