Question: One generalization of the binomial distribution is to allow the success probability to vary according to a distribution. A standard model for this situation is \(X \mid P \sim \operatorname{BIN}(n, p)\) and \(P \sim \operatorname{Beta}(\alpha, \beta)\)

1. Find E[X].
2. Show that \(\operatorname{Var}[X]=n E[P(1-E[P])]+n(n-1) \operatorname{Var}[P]\). The first term reflects binomial variation with success probability E[P], and the second term is often called "extra binomial" variation, showing how this model has a variance that is higher than the binomial alone. Additionally, explain in plain language why you intuitively would expect this model to have a larger variance than the regular binomial based on what you know about variance and what it represents.
3. Show that the marginal distribution of $X$ is given by the Beta-Binomial distribution:
   \[P(X=x)=\left( \begin{array}{*{35}{l}} n \\ x \\ \end{array} \right)\frac{\Gamma (\alpha +\beta )}{\Gamma (\alpha )\Gamma (\beta )}\cdot \frac{\Gamma (x+\alpha )\Gamma (n-x+\beta )}{\Gamma (\alpha +\beta +n)}\]
4. A variation of this model is when \(Y \mid P \sim \operatorname{NB}(r, P)\) and \(P \sim \operatorname{Beta}(\alpha, \beta)\). Find the marginal distribution of \(Y\).

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