Problems

A hostile naval power possesses a large, unknown number \(N\) of submarines. Interception of radio signals yields a small number \(n\) of their identification numbers \(X_i\) (\(i=1,2,...,n\)), which are taken to be independent and uniformly distributed over the continuous range from \(0\) to \(N\). Show that \(Z_1\) and \(Z_2\), defined by $$ Z_1 = {n+1\over n} {\max}\{X_1,X_2,...,X_n\} \hspace{0.3in} {\rm and} \hspace{0.3in} Z_2 = {2\over n} \sum_{i=1}^n X_i \;, $$ both have means equal to \(N\). Calculate the variance of \(Z_1\) and of \(Z_2\). Which estimator do you prefer, and why?