Consider a random sample of size \(n\) from the unit Pareto distribution with PDF

\[ f(x)=2 x^{-2} \quad \text { for } 0 \leq x \]

that is, for the Pareto with \(\alpha=2\) and \(\gamma=1\).

(a) Write the cumulative distribution function for the maximum order statistic \(X_{(n: n)}\) in the sample.

(b) Now form the normalized sequence \(Z_{n}=c_{n} X_{(n: n)}+d_{n}\), where \(\left\{c_{n}\right\}\) is the sequence \(\{1 / n\}\) and \(\left\{d_{n}\right\}\) is the constant sequence \(\{0\}\), and write the CDF of \(Z_{n}\).

(c) Now take the limit as \(n \rightarrow \infty\) of the CDF of \(Z_{n}\).
This is the CDF of an extreme value distribution, and is in the form of the generalized extreme value distribution CDF given in equation (3.110).

\(F_G(x)= \begin{cases}\mathrm{e}^{-(1+\xi x)^{-1 / t}}, & \text { for } \xi eq 0 \\ \mathrm{e}^{-\mathrm{e}^{-x}}\end{cases} \tag{3.110}\)