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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?

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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)

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    $\begingroup$ You could also use Mean[pmd] $\endgroup$
    – Bob Hanlon
    Dec 21, 2022 at 5:30
  • $\begingroup$ @BobHanlon yes would have been shorter. Thank you. $\endgroup$
    – ubpdqn
    Dec 21, 2022 at 7:22

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