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Having trouble understanding something I read in a paper recently. Say we have $X \sim \operatorname{Binomial}(N,p).$ The paper states:

$$E[X \mid N,p] = Np$$ (so far so good)

and

$$E[X] = \mu p$$

Where is $p$ in the second line coming from, if $\mu$ is defined as $Np$ for $\operatorname{Binomial}(N,p)$?

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    $\begingroup$ The first notation suggests both $N$ and $p$ are random variables, so perhaps $\mu$ is related to them. Since "$\mu$" has no inherent or commonly understood meaning, surely it is defined or described elsewhere in that paper. I think we need a more complete statement of the context of that paper in order to understand your question. $\endgroup$
    – whuber ♦
    Commented Jul 1 at 17:28

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It seems that what must have been meant was $$ X\mid N \sim \operatorname{Binomial}(N,p) $$ and $\mu=\operatorname E(N).$

In that case you have $\operatorname E(X\mid N) = Np$ and then $$ \operatorname E(X) = \operatorname E(\operatorname E(X\mid N)) = \operatorname E(Np) = p\operatorname E(N) = p\mu. $$

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