Let (left{X_{n}ight}) be a sequence of independent and identically distributed random variables where the distribution function of
Question:
Let \(\left\{X_{n}ight\}\) be a sequence of independent and identically distributed random variables where the distribution function of \(X_{n}\) is
\[F_{n}(x)= \begin{cases}1-x^{-\theta} & \text { for } x \in(1, \infty) \\ 0 & \text { for } x \in(-\infty, 1]\end{cases}\]
Define a new sequence of random variables given by
\[Y_{n}=n^{-1 / \theta} \max \left\{X_{1}, \ldots, X_{n}ight\}\]
for all \(n \in \mathbb{N}\).
a. Prove that the distribution function of \(Y_{n}\) is \[G_{n}(y)= \begin{cases}{\left[1-\left(n x^{\theta}ight)^{-1}ight]^{n}} & x>1 \\ 0 & \text { for } x \leq 1\end{cases}\]
b. Consider the distribution function of a random variable \(Y\) given by \[G(y)= \begin{cases}\exp \left(-x^{-\theta}ight) & x>1 \\ 0 & \text { for } x \leq 1\end{cases}\]
Prove that \(G_{n} \leadsto G\) as \(n ightarrow \infty\).
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