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% File Name: LimitingDistributionCLTproblem.tex
% Submitted: Dr. Eric Suess,Statistics 6401, Fall 2012
% Submitted Date: January 1st, 2002
% Accepted Date: January 9th, 2002
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\begin{document}
\pagestyle{myheadings} \markright{Statistics 6401, Winter 2012}

\large
\begin{center}
\textbf{Limiting Distribution and CLT Problems}
\end{center}


\begin{enumerate}
  \item Let $X_1,X_2,\ldots,X_n$ be a random sample from
  $Exp(\lambda)$. Let $Y_1$=min\{$X_1,X_2,\ldots,X_n$\}. Find the
  limiting distribution of $Y_1$.
  \item Let $X_1,X_2,\ldots,X_n$ be a random sample from
  $F(x)=1-\frac{1}{x}, x\geq1$.
  \begin{enumerate}
    \item Find the c.d.f. of $Y_1$=min\{$X_1,X_2,\ldots,X_n$\}.
    \item Let $Z_n=n(1-F_X (Y_n))$ where
    $Y_n$=max\{$X_1,X_2,\ldots,X_n$\},
    find \[ \lim_{n \rightarrow \infty} F_{Z_{n}}(t).\]
  \end{enumerate}
  \item Let $X_1,X_2,\ldots,X_n$ be independent \large $\chi^2(1)$ \normalsize random
  variables.
  Then \[ Y_{n} = \sum_{i=1}^{n} X_{i} \sim \large \chi^2(n)
  \normalsize.\]

  The C.L.T. says \large $Z_n=\frac{Y_n - n}{\sqrt{2n}} \stackrel{d}{\longrightarrow} Z \sim N(0,1)$ \normalsize.

  Show that \large $W_n = \sqrt{2Y_n} - \sqrt{2n-1}
  \stackrel{d}{\longrightarrow}Z \sim N(0,1)$ \normalsize.
\end{enumerate}

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