Binomial distribution where probability parameter is also a distribution

Question

Suppose I have $J \sim$ Bin$(25, P)$, but instead of having fixed $P$, I have a known PDF for continuous $P$, given by $f_{P}(p)$. How would I find $\mathbb{E}[J]$, given this to be the case?

Answer 1

You can use ParameterMixtureDistribution(https://reference.wolfram.com/language/ref/ParameterMixtureDistribution.html)

For example:

pmd = ParameterMixtureDistribution[BinomialDistribution[25, p], 
   p \[Distributed] BetaDistribution[q, b]];
Expectation[x, x \[Distributed] pmd]

yields: (25 q)/(b + q)