<s>[INST] How do I show that $Var(Y) = n\frac{(\theta _{1} - \theta _{2})^{2}}{(n+1)^{2} (n+2)}$? My original pdf is $f(y) = \frac{n (y_{n} - \theta_{1})^{n-1}}{(\theta_{2} - \theta_{1})^{n}}$ for $\theta_{1} < y < \theta_{2}$.
After using U-substitution, I obtain $E(Y) = \frac{n \theta_{2} + \theta_{1}}{(n+1)}$. 
For variance of $Y$, I need to find $E(Y^2)$ first. 
This is what I have with U-substitution: $E(Y^2) = \frac{n\theta_{2} - n \theta_{1} + 2n \theta_{1} \theta_{2} - n\theta_{1}^{2} + \theta_{1}^{2}}{(n+1)}$.
But with this the $Var(Y)= E(Y^2) - \left [E(Y)  \right ]^{2} \neq n\frac{(\theta _{1} - \theta _{2})^{2}}{(n+1)^{2} (n+2)}$
My mistake is definitely the $E(Y^2)$ part, so I would appreciate it if someone here can go over it with me. 
Edit:
$$\begin{eqnarray}
   \mathbb{E}\left(Y^2\right) &=& \mathbb{E}\left(\left(Y-\theta_1\right)^2\right) + 2 \theta_1 \cdot \mathbb{E}\left(\left(Y-\theta_1\right)\right) + \theta_1^2 \\ &=& \frac{n}{n+2} \left(\theta_2-\theta_1\right)^2 + 2 \theta_1 \cdot \frac{n}{n+1} \left(\theta_2-\theta_1\right) + \theta_1^2
\end{eqnarray}$$ from below and also my own work.
But now how do I simplify $$Var(Y) =  \mathbb{E}\left(Y^2\right) - \left [ \mathbb{E}\left(Y\right)  \right ]^{2}$$ to  $$Var(Y) = n\frac{(\theta _{1} - \theta _{2})^{2}}{(n+1)^{2} (n+2)}?$$
 [/INST] The conceptual way to do this is to recognize that $Y$ is a location-scale transformation of a beta distribution.  If $f_Y(y) \propto (y-\theta_1)^{n-1}$ for $y \in [\theta_1, \theta_2]$, then this suggests the linear, order-preserving transformation $$X = g(Y) = \frac{Y-\theta_1}{\theta_2 - \theta_1}.$$  Then $Y = g^{-1}(X) = (\theta_2 - \theta_1)X + \theta_1$, and the PDF of $X$ is given by $$f_X(x) = f_Y(g^{-1}(x)) \left| \frac{dg^{-1}}{dx} \right| = nx^{n-1}, \quad x \in [0,1].$$  We immediately recognize that $X \sim \mathrm{Beta}(a = n, b = 1)$.  Hence $$\begin{align*} \mathrm{E}[X] &= \frac{a}{a+b} = \frac{n}{n+1}, \\  \mathrm{Var}[X] &= \frac{ab}{(a+b)^2(a+b+1)} = \frac{n}{(n+1)^2(n+2)}. \end{align*}$$  Since $Y = (\theta_2 - \theta_1)X + \theta_1$, it immediately follows that $\mathrm{Var}[Y] = (\theta_2-\theta_1)^2 \mathrm{Var}[X]$ and the result is proven.
In the event that we wish to derive the variance of $X$ directly from its PDF, the computation is a trivial exercise in integration:  $$\mathrm{E}[X^k] = \int_{x=0}^1 x^k nx^{n-1} \, dx = n \int_{x=0}^1 x^{n+k-1} \, dx = n \left[\frac{x^{n+k}}{n+k} \right]_{x=0}^1 = \frac{n}{n+k}.$$  Thus $$\mathrm{Var}[X] = \mathrm{E}[X^2] - \mathrm{E}[X]^2 = \frac{n}{n+2} - \Bigl(\frac{n}{n+1}\Bigr)^2 = \frac{n}{(n+1)^2(n+2)}.$$
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