3.13 Let \( X \sim \operatorname{Bin}(n, \theta) \), and consider a conjugate \( \operatorname{Beta}(a, b) \) prior distribution for \( \theta \), as in Example 3.1. Show that if we reparametrise from \( (a, b) \) to \( (\mu, M) \), where \( \mu= \) \( a /(a+b) \) and \( M=a+b \), the marginal distribution of \( X \) is of beta-binomial form:
\[
\operatorname{Pr}(X=x \mid \mu, M)=\frac{\Gamma(M)}{\Gamma(M \mu) \Gamma\{M(1-\mu)\}}\left(\begin{array}{l}
n \\
x
\end{array}\right) \frac{\Gamma(x+M \mu) \Gamma\{n-x+M(1-\mu)\}}{\Gamma(n+M)}
\]
