A statistician wants to use iid outcomes from some exponential distribution $f(x;\theta )=\left(1/\theta \right)e^{-x/\theta }{I}_{(0,\infty )}\left(x\right)$ to generate an estimate of the variance of the exponential density, ${\theta }^2$. She wants to use the outcome of ${\overline{X}}_n^2$ where ${\overline{X}}_n=n^{-1}{\sum }_{i=1}^n{X}_i$ to generate an estimate of ${\theta }^2$.

(a) Does $E\left({\overline{X}}_n^2\right)={\theta }^2$ ? Does ${lim}_{n\to \infty }E\left({\overline{X}}_n^2\right)={\theta }^2$ ?

(b) Does plim $\left({\overline{X}}_n^2\right)={\theta }^2$ ?

(c) Define an asymptotic distribution for ${\overline{X}}_n^2$.