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I have the following compound binomial random variable:

$B_n \sim \operatorname{Binomial}(X_n, 1-p)$, where $X_n$ is itself another random variable.

This means that $(B_n \mid X_n = x_n) \sim\operatorname{Binomial} (x_n, 1-p)$

I understand that $E(B_n)$ = $E(E(B_n | X_n))$

= $E((1-p)*X_n)$ = $(1-p) * E(X_n)$

And then I can calculate $E(X_n)$ based on whatever the distribution of the random variable $X_n$ is.

Now, I am trying to calculate the generating function of $B_n$:

$E(t^{B_n})$

But I am getting stuck in the way the Binomial random variable is actually $B_n$ | $X_n$

(so just the unconditional $B_n$ is not Binomial)

Given this, what other way can I use to calculate $E(t^{B_n})$ ? Thanks for any help.

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If you know the generating function of a Binomial random variable, then you get that $$E(t^{B_n} \, | \, X_n) = (p+(1-p)t)^{X_n}$$ If you know the distribution of $X_n$. then you may use the law of total expectation to get: $$E(t^{B_n}) = E[E(t^{B_n} \, | \, X_n)] = E\left[ (p+(1-p)t)^{X_n} \right]$$

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  • $\begingroup$ Thank you! This is helpful. I'll work it out further now. $\endgroup$ Oct 16, 2023 at 2:52

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