Suppose $\displaystyle X$ is exponentially distributed with density $\displaystyle f_{X}(x) = e^{-x}1_{x>0}.$ Consider $\displaystyle Z = \frac{h}{X},$ where $\displaystyle h > 0.$ Our goal is to estimate $\displaystyle h.$

(a) Find the density function of $\displaystyle Z.$

(b) Let $\displaystyle Z1,...,Zn$ be an i.i.d. sample from $\displaystyle f_{Z}(z),$ the density of $\displaystyle Z.$ Find the MLE of $\displaystyle h,$ denoted by $\displaystyle \hat{h}.$

(c) Find the asymptotic variance of $\displaystyle \hat{h}.$