2. The random variable X has the following probability density function:
\begin{equation*}
f(x) = \frac{2}{\lambda^2}xe^{-x^2/\lambda^2}, \quad x \ge 0
\end{equation*}
where $\lambda$ is a positive parameter.
(15 points)
(a) It can be shown that
\begin{equation*}
E(X) = \frac{\sqrt{\pi\lambda}}{2}
\end{equation*}and
\begin{equation*}
E(X^2) = \lambda^2
\end{equation*}
Set up (but do not evaluate) the definite integrals that are needed to show this.
(b) Let $X_1, X_2, ..., X_n$ be a random sample from X. Find an unbiased estimator
$\hat{\lambda}$ for $\lambda$ of the form
(some constant) $\cdot \overline{X}$
(c) Find V($\hat{\lambda}$).
(d) Is $\hat{\lambda}$ a consistent estimator for $\lambda$? Explain.
