Given that Y has a binomial distribution with parameters $n$ and $P$, but that $P$ varies from day to day according to a beta distribution with parameters $\alpha$ and $\beta$, I'm trying to show that $$E[Y]=\frac{n \alpha}{\alpha + \beta}$$ But I'm not sure how to. I understand that $E[Y]=nP$, and that the expectation of the beta distribution is $\frac{1}{1+\frac{\beta}{\alpha}} = \frac{\alpha}{\alpha + \beta}$ So does that mean that I can just substitute the expectation of the beta distribution as $P$ to prove it? Is this the right way of proving it? Or does this have anything to do with conditional expectations?
Thank you.
You have $\mathbb E[Y]=\mathbb E[nP]$ (do you have to prove this or can you take it as given?)
and $n$ is constant so $\mathbb E[Y]=n \mathbb E[P]$
and $\mathbb E[P] = \dfrac{\alpha}{\alpha + \beta}$ (do you have to prove this or can you take it as given?)
so $\mathbb E[Y] = \dfrac{n\alpha}{\alpha + \beta}$
essentially as you said, and using the fact $n$ is a multiplicative constant. It would be different if $n$ varied and was not independent of $P$ or $\alpha$ or $\beta$