4: Pareto distribution Let $X_{1}, \ldots, X_{n} $ be i.i.d. from a Pareto $(\alpha, 3)$ distribution where $\alpha$ is unknown, that is $$ f_{X} (x \mid \alpha)=\left\{\begin{array}{11} \frac{3 \alpha^{3}}{x^{4}}, & x \geq \alpha 0, & \text { otherwise } \end{array} \quad \text { where } \alpha>0 . ight. $$ c) Can show $E\left[X_{(1)} ight ]=\frac{3 n}{3 n-1} \cdot \alpha$, and $\operatorname{Var}\left[X_{(1)} ight)=\frac{3 n}{(3 n-1)^{2}(3 n-2)} \cdot \alpha^{2}$, where $X_{(1)}=\min \left\{X_{i} ight\}_{i=1}^{n} $ Use this to modify $\hat{\alpha}_{m 1 e) $ such that the resulting estimator $\hat{\alpha}_{u b}$ is unbiased. Also find the variance of $\hat{\alpha}_{u b}$. SP.JG.028