<s>[INST] Finding variance from a beta distribution Suppose $[X\mid \theta]\sim\operatorname{Uniform}([0, \theta])$ and $B$ is is a beta distributed random variable.  Additionally, $\max(X_1,…,X_n)/θ\sim \operatorname{Beta}(n+1,1)$
Show that $\hat{θ}= \frac{n+2}{n+1} \max(X_1,…,X_n)$ and find the variance of $\hat\theta$ given $\theta$.

I found: $E[B]= \frac{n+1}{n+2}$ and $V[B]= \frac{n+1}{(n+2)^2(n+3)}$
How do I find the variance of $\hat{\theta}\mid \theta?$
 [/INST] I will assume you meant $B\sim\operatorname{Beta}(n+1,1).$
Recall that $\operatorname{var}(B) = \operatorname E\big(B^2\big) - \big( \operatorname E(B) \big)^2$.
\begin{align}
& \operatorname E(B^2) = \int_0^1 x^2\cdot f_B(x)\,dx \\[8pt]
= {} & \int_0^1 x^2\cdot \frac{\Gamma(n+2)}{\Gamma(n+1)\Gamma(1)} \cdot x^n\,dx \\[8pt]
= {} & (n+1) \int_0^1 x^{n+2}\,dx = \frac{n+1}{n+3}.
\end{align}
So the variance is
$$
\frac{n+1}{n+3} - \left( \frac{n+1}{n+2} \right)^2 = \frac{(n+1)(n+2)^2 - (n+1)^2}{(n+3)(n+2)^2} = \frac{n+1}{(n+3)(n+2)^2}.
$$
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