Question: Let \({{Y}_{1}},....,{{Y}_{n}}\sim Exp\left( \theta \right)\), so

\[f\left( y \right)=\frac{1}{\theta }{{e}^{-y/\theta }},\,\,y>0,\theta >0\]

(a) Find the distribution of the smallest order statistics \(Y\left( 1 \right)=\underset{1\le i\le n}{\mathop{\min }}\,{{Y}_{i}}\).

(b) Calculate \(E\left( Y\left( 1 \right) \right)\) and \(\operatorname{var}\left( Y\left( 1 \right) \right)\)

(c)Let \({{\hat{\theta }}_{1}}=cY\left( 1 \right)\), for some constant c. Find a value of c such that \(E\left( {{{\hat{\theta }}}_{1}} \right)=\theta \)

(d) Let \({{\hat{\theta }}_{2}}=\bar{Y}\), where \(\bar{Y}=\frac{1}{n}\left( {{Y}_{1}}+...+{{Y}_{n}} \right)\). Calculate \(E\left( {{{\hat{\theta }}}_{2}} \right)\) and \(\operatorname{var}\left( {{{\hat{\theta }}}_{2}} \right)\)

(e) The goal is to find the estimator with the minimum mean square error (MSE). Calculate and compare the MSE of the two estimators \({{\hat{\theta }}_{1}}\) and \({{\hat{\theta }}_{2}}\) from parts (c) and (d).