Binomial distribution where the number of experiments is binomially distributed

Question

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.

Answer 1

Using the law of total covariance, \begin{align} \operatorname{Cov}(K,N) &=E\operatorname{Cov}(K,N|N)+\operatorname{Cov}(EK|N,EN|N) \\&=E 0+\operatorname{Cov}(pN,N) \\&=0+p\operatorname{Var}(N) \\&=pmq(1-q). \end{align}