Covariance of two r.v. $X\sim B(Z,\alpha)$, $Y\sim B(X\alpha,\delta)$

Question

Suppose $Z_i$ are i.i.d. random variable and $Z_i\alpha$ are positive integers.

For the following two random variables $X$ and $Y$, I would like to compute the $\mathrm{Cov}(X,Y)$ where $X \sim B(Z,\alpha)$, $Y\sim B(X,\delta)$

Any inputs or comments would be appreciated.

Answer 1

I would like to compute the $Cov(X,Y)$ where $X∼B(Z,α)$ and $[Y\mid X]\sim B(X,δ)$

… or so I interpret your question. That is $X$ is the count of $Z$ independent Bernoulli trials with identical success rate $\alpha$, and conditioned on a given $X$, that $Y$ is the count of $X$ independent Bernoulli trials with identical success rate $\delta$.

Then by treating $Z, \alpha$, and $\delta$ as constant terms:

$$\begin{align}\mathsf {Cov}(X,Y) &=\mathsf E(XY)-\mathsf E(X)~\mathsf E(Y)&&\text{by definition}\\ &=\mathsf E(X~\mathsf E(Y\mid X))-\mathsf E(X)~\mathsf E(\mathsf E(Y\mid X))&&\text{by Law of Total Expectation}\\&~~\vdots\end{align}$$

Everything else is just knowing what is the expectation for a Binomially Distributed Random Variable in terms of its trial amount and success rate parameters.