Unbiased Estimator for Uniform Distribution

Question

$X_1$ , a sample size 1 is drawn from a uniform distribution over $[0,\theta]$. Find an unbiased estimator for the variance of the population. Find a function for $X_1$, $\tau(X_1)$ such that $E(\tau(X_1))=\theta^2/12$.

I know the expected value of this uniform distribution can be found $$\int_{0}^{\theta} x(\frac{1}{\theta}) dx$$. Would I just be looking for some x such that $$\int_{0}^{\theta} x(\frac{1}{\theta}) dx=E(\tau(X_1))=\theta^2/12$$

Answer 1

You're not looking for some $x$ that satisfies that equation at the end. Indeed, $x$ is a dummy variable in the equation (you could write $u$ for $x$ everywhere in the integral without altering the value), so it won't even appear in the equation after you evaluate the integral.

The question simply seeking a function of $X_1$ whose expectation is the variance of the distribution, which is $\theta^2/12$.

By inspection, $E(X_1)=\theta/2$, so no linear function of $X_1$ would do. The obvious "guess" at a functional form is a multiple of $X_1^2$. So evaluate $E(X_1^2)$ and see if you get some multiple of $\theta^2$, and if you do, then see what you'd multiply it by to get the correct variance.