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\mode<presentation>

\title{Limit Theorems\footnote{ This slide show is an open-source document. See last slide for copyright information.}}
\subtitle{STA 256: Fall 2018}
\date{} % To suppress date

\begin{document}

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\begin{frame}
  \titlepage
\end{frame}

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\begin{frame}
\frametitle{Overview}
\tableofcontents
\end{frame}

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\section{Law of Large Numbers}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Infinite Sequence of random variables} 
%\framesubtitle{} 
{\Large $T_1, T_2, \ldots$ } \pause \vspace{5mm}

\begin{itemize}
    \item We are interested in what happens to $T_n$ as $n \rightarrow \infty$. \pause
    \item Why even think about this? \pause
    \item For fun. \pause
    \item And because $T_n$ could be a sequence of \emph{statistics}\pause, numbers computed from  sample data. \pause
    \item For example, $T_n = \overline{X}_n \pause = \frac{1}{n}\sum_{i=1}^nX_i$. \pause
    \item $n$ is the sample size. \pause
    \item $n \rightarrow \infty$ is an approximation of what happens for large samples. \pause
    \item Good things should happen when estimates are based on more information.
\end{itemize}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Convergence}
%\framesubtitle{}
\begin{itemize}
    \item Convergence of $T_n$ as $n \rightarrow \infty$ is not an ordinary limit, because probability is involved. \pause
    \item There are several different types of convergence. \pause
    \item In this class, we will work with \emph{convergence in probability} and \emph{convergence in distribution}.
\end{itemize}
\end{frame}

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\begin{frame}
\frametitle{Convergence in Probability}
%\framesubtitle{}
Definition: The sequence of random variables $T_1, T_2, \ldots$ is said to converge in probability to the constant $c$ if for all $\epsilon > 0$,

{\LARGE
\begin{displaymath}
    \lim_{n \rightarrow \infty}P\{|T_n-c|\geq\epsilon\} = 0
\end{displaymath} \pause
} % End size
% Or equivalently, \pause
% \begin{displaymath}
%     \lim_{n \rightarrow \infty}P\{|T_n-c|\leq\epsilon\} = 1
% \end{displaymath}  \pause
Observe
\begin{eqnarray*}
    |T_n-c| < \epsilon \pause & \Leftrightarrow & -\epsilon < T_n-c < \epsilon \\ \pause
    & \Leftrightarrow & c-\epsilon < T_n < c+\epsilon \\ \pause
\end{eqnarray*} 
\begin{picture}(10,10)
    % Line, direction (1,0), horizontal extent 200, starting point (50,0)
    \put(50,0){\line(1,0){200} } 
    \put(150,5){\line(0,-1){10} }
    \put(148,-15){$c$}
    \put(100,-2){(} % Left parenthesis
    \put(200,-2){)} % Right parenthesis
    \put(90,-15){$c-\epsilon$}
    \put(190,-15){$c+\epsilon$}
\end{picture}
\end{frame}

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\begin{frame}
\frametitle{Example: $T_n \sim U(-\frac{1}{n}, \frac{1}{n})$}
\framesubtitle{Convergence in probability means $\lim_{n \rightarrow \infty}P\{|T_n-c|\geq\epsilon\} = 0$}
\begin{picture}(10,10)(25,-25)
    % Line, direction (1,0), horizontal extent 200, starting point (50,0)
    \put(50,0){\line(1,0){200} } 
    \put(150,5){\line(0,-1){10} }
    \put(148,-15){$c$}
    \put(100,-2){(} % Left parenthesis
    \put(200,-2){)} % Right parenthesis
    \put(90,-15){$c-\epsilon$}
    \put(190,-15){$c+\epsilon$}
\end{picture}  \pause
\begin{itemize}
    \item $T_1$ is uniform on $(-1,1)$. \pause Height of the density is $\frac{1}{2}$. \pause
    \item $T_2$ is uniform on $(-\frac{1}{2},\frac{1}{2})$.  \pause Height of the density is 1. \pause
    \item $T_3$ is uniform on $(-\frac{1}{3},\frac{1}{3})$.  \pause Height of the density is $\frac{3}{2}$. \pause
    \item Eventually, $\frac{1}{n} < \epsilon$ \pause and $P\{|T_n-0|\geq\epsilon\} = 0$\pause, forever. \pause
    \item Eventually means for all $n>\frac{1}{\epsilon}$. 
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Example: $X_1, \ldots, X_n$ are independent $U(0,\theta)$}
\framesubtitle{Convergence in probability means $\lim_{n \rightarrow \infty}P\{|T_n-c|\geq\epsilon\} = 0$} \pause
For $0 < x < \theta$, \pause
\begin{itemize}
    \item[] $F_{x_{i}}(x) = \int_0^x \frac{1}{\theta} \, dx \pause = \frac{x}{\theta}$. \pause 
    \item[] $Y_n = \max_i (X_i)$. \pause 
    \item[] $F_{y_n}(y) = \left(\frac{x}{\theta}\right)^n$ \pause 
\end{itemize} \vspace{2mm}

\begin{picture}(10,10) % (25,-25)
    % Line, direction (1,0), horizontal extent 200, starting point (50,0)
    \put(50,0){\line(1,0){200} } 
    \put(150,5){\line(0,-1){10} }
    \put(148,-15){$\theta$}
    \put(100,-2){(} % Left parenthesis
    \put(200,-2){)} % Right parenthesis
    \put(90,-15){$\theta-\epsilon$}
    \put(190,-15){$\theta+\epsilon$}
\end{picture}  \pause \vspace{5mm}

\begin{eqnarray*}
    P\{|Y_n-\theta|\geq\epsilon\} & = & F_{y_n}(\theta-\epsilon) \\ \pause
    & = & \left(\frac{\theta-\epsilon}{\theta}\right)^n \\ \pause 
    & \rightarrow & 0 \pause \mbox{ ~~~because } \frac{\theta-\epsilon}{\theta}<1. \pause
\end{eqnarray*}
So the observed maximum data value goes in probability to $\theta$, the theoretical maximum data value.
\end{frame}

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\begin{frame}
\frametitle{The Law of Large Numbers} \pause
%\framesubtitle{} 
Theorem: Let $X_1, \ldots, X_n$ be independent random variables with expected value $\mu$ and variance $\sigma^2$. \pause Then $\overline{X}_n = \frac{1}{n}\sum_{i=1}^nX_i$ converges in probability \pause to $\mu$. \pause \vspace{3mm}
{ \small
\begin{itemize}
    \item This is not surprising, because $E(\overline{X}_n) = \mu$ and \pause
    \item $Var(\overline{X}_n) = \frac{\sigma^2}{n}$ \pause
\begin{eqnarray*}
    Var\left( \frac{1}{n}\sum_{i=1}^nX_i \right)  \pause 
    & = & \frac{1}{n^2} Var\left(\sum_{i=1}^nX_i \right)  \\ \pause 
    & = & \frac{1}{n^2} \sum_{i=1}^nVar(X_i) \\ \pause 
    & = & \frac{1}{n^2} \sum_{i=1}^n \sigma^2 \pause = \frac{1}{n^2} n\sigma^2 \pause
          = \frac{\sigma^2}{n} \pause \downarrow 0.
\end{eqnarray*}
    \item And the implications are huge. 
\end{itemize}
} % End size
\end{frame}


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\begin{frame}
\frametitle{Probability is long-run relative frequency} \pause
%\framesubtitle{Implied by the Law of Large Numbers} 
This follows from the Law of Large Numbers. \pause 

Repeat some process over and over a lot of times, and count how many times the event $A$ occurs. \pause Independently for $i=1, \ldots, n$, \pause
\begin{itemize}
    \item Let $X_i(\omega)=1$ if $\omega \in A$, \pause and $X_i(\omega)=0$ if $\omega \notin A$. \pause
    \item So $X_i$ is an \emph{indicator} for the event $A$. \pause
    \item $X_i$ is Bernoulli, with $P(X_i=1) = p \pause = P(A)$. \pause
    \item $E(X_i) = \sum_{x=0}^1 x \, p(x) \pause = 0\cdot(1-p) + 1\cdot p \pause
           =p$. \pause
    \item $\overline{X}_n$ is the proportion of times the event occurs in $n$ independent trials. \pause
    \item The proportion of successes converges in probability to $P(A)$. % \pause 
%    \item So while $\overline{X}_n$ is a random quantity with its own probability distribution, \pause
%    \item That distribution shrinks to fit in a tiny interval around $P(A)$, no matter how small the interval.
\end{itemize} \vspace{3mm}
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    % Line, direction (1,0), horizontal extent 200, starting point (50,0)
    \put(50,0){\line(1,0){200} } 
    \put(150,5){\line(0,-1){10} }
    \put(148,-15){$p$}
    \put(100,-2){(} % Left parenthesis
    \put(200,-2){)} % Right parenthesis
    \put(90,-15){$p-\epsilon$}
    \put(190,-15){$p+\epsilon$}
\end{picture}

\end{frame}

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\begin{frame}
\frametitle{Proof of the Law of Large Numbers}
\framesubtitle{Using $E(\overline{X}_n)= \mu$ and $Var(\overline{X}_n) = \frac{\sigma^2}{n}$} \pause 
\begin{itemize}
    \item Chebyshev's inequality says $P(|X-\mu| \geq k\sigma) \leq \frac{1}{k^2}$ \pause
    \item Here, $X$ is replaced by $\overline{X}_n$  and $\sigma$ is replaced by $\frac{\sigma}{\sqrt{n}}$. \pause
    \item So Chebyshev's inequality becomes 
    $P(|\overline{X}_n-\mu| \geq k\frac{\sigma}{\sqrt{n}}) \leq \frac{1}{k^2}$. \pause
    \item $k>0$ is arbitrary, so set $\frac{k\sigma}{\sqrt{n}}=\epsilon$. \pause
    \item Then $k=\frac{\epsilon\sqrt{n}}{\sigma}$ \pause and $\frac{1}{k^2}= \frac{\sigma^2}{\epsilon^2 n}$. \pause
    \item Thus, \pause
\end{itemize}
{\LARGE
\begin{displaymath}
    0 \leq P\{|\overline{X}_n-\mu|\geq\epsilon\} \pause \leq \frac{\sigma^2}{\epsilon^2 n} \pause 
    \downarrow 0 \
\end{displaymath} 
} % End size
Squeeze. $\blacksquare$
\end{frame}

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\begin{frame}
\frametitle{Theorem}
\framesubtitle{Proof omitted in 2018} \pause
Let $g(x)$ be a function that is continuous at $x=c$. If $T_n$ converges in probability to $c$, then $g(T_n)$  converges in probability to $g(c)$. \pause \vspace{5mm}
% 

Examples:
\begin{itemize}
     \item A Geometric distribution has expected value $1/p$. \pause $1/\overline{X}_n$ converges in probability to $1/E(X_i) \pause = p$. \pause
     \item  A Uniform($0,\theta$) distribution has expected value $\theta/2$. \pause $2\overline{X}_n$ converges in probability to $2E(X_i) \pause = 2\frac{\theta}{2}=\theta$.
\end{itemize}


\end{frame}


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\section{Central Limit Theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Convergence in distribution}
\framesubtitle{Another mode of convergence}  \pause
Definition: Let the random variables $X_1, X_2 \ldots$ have cumulative distribution functions $F_1(x), F_2(x) \ldots$\pause, and let the random variable $X$ have cumulative distribution function $F(x)$\pause. The (sequence of) random variable $X_n$ is said to \emph{converge in distribution} to $X$ if \pause

{\LARGE
\begin{displaymath}
    \lim_{n \rightarrow \infty}F_n(x) = F(x)
\end{displaymath} \pause \vspace{4mm}
} % End size

at every point where $F(x)$ is continuous. 
\end{frame}

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\begin{frame}
\frametitle{Example: Convergence to a Bernoulli with $p=\frac{1}{2}$}
\framesubtitle{$\lim_{n \rightarrow \infty}F_n(x) = F(x)$ at all continuity points of $F(x)$}  \pause
\begin{displaymath}
    p_n(x) = 
     \left\{ \begin{array}{cl} % ll means left left
   1/2 & \mbox{for } x=\frac{1}{n}  \\
   1/2 & \mbox{for } x=1+\frac{1}{n}  \\
   0           & \mbox{Otherwise}
   \end{array}  \right.
\end{displaymath} \vspace{3mm} \pause

\begin{picture}(10,10)(0,-10)
    \put(15,-2){$n=1$}
    \put(50,0){\line(1,0){200} } 
    \put(150,5){\line(0,-1){10} }
    \put(100,5){\line(0,-1){10} }
    \put(200,5){\line(0,-1){10} }
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    \put(148,-15){1}
    \put(198,-15){2}
    \put(197.5,-2){$\bullet$}
    \put(147.5,-2){$\bullet$}
\end{picture} \pause

\begin{picture}(10,10)(0,10)
    \put(15,-2){$n=2$}
    \put(50,0){\line(1,0){200} } 
    \put(150,5){\line(0,-1){10} }
    \put(100,5){\line(0,-1){10} }
    \put(200,5){\line(0,-1){10} }
    \put(98,-15){0}
    \put(148,-15){1}
    \put(198,-15){2}
    \put(172.5,-2){$\bullet$}
    \put(122.5,-2){$\bullet$}
\end{picture} \pause

\begin{picture}(10,10)(0,30)
    \put(15,-2){$n=3$}
    \put(50,0){\line(1,0){200} } 
    \put(150,5){\line(0,-1){10} }
    \put(100,5){\line(0,-1){10} }
    \put(200,5){\line(0,-1){10} }
    \put(98,-15){0}
    \put(148,-15){1}
    \put(198,-15){2}
    \put(164.7,-2){$\bullet$}
    \put(114.7,-2){$\bullet$}
\end{picture} \pause \vspace{15mm}


\begin{itemize}
    \item For $x<0$, $\lim_{n \rightarrow \infty}F_n(x)=$ \pause $0$ \pause
    \item For $0<x<1$, $\lim_{n \rightarrow \infty}F_n(x)=$ \pause $\frac{1}{2}$ \pause
    \item For $x>1$, $\lim_{n \rightarrow \infty}F_n(x)=$ \pause $1$ \pause
    \item What happens at $x=0$ and $x=1$ does not matter.
\end{itemize}
\end{frame}

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\begin{frame}
\frametitle{Convergence to a constant} \pause
%\framesubtitle{} 
{\small
Consider a ``degenerate" random variable $X$ with $P(X=c)=1$. \pause \vspace{3mm}

\begin{picture}(10,10) % (25,-25)
    % Line, direction (1,0), horizontal extent 200, starting point (50,0)
    \put(50,0){\line(1,0){200} } 
    \put(150,5){\line(0,-1){10} }
    \put(148,-15){$c$}
    \put(100,-2){(} % Left parenthesis
    \put(200,-2){)} % Right parenthesis
    \put(90,-15){$c-\epsilon$}
    \put(190,-15){$c+\epsilon$}
\end{picture}  \pause \vspace{5mm}

Suppose $X_n$ converges in probability to $c$. \pause
\begin{itemize}
    \item Then for any $x>c$, $F_n(x) \rightarrow 1$ for $\epsilon$ small enough. \pause
    \item And for any $x<c$, $F_n(x) \rightarrow 0$ for $\epsilon$ small enough. \pause
    \item So $X_n$ converges in distribution to $c$.
\end{itemize} \pause
Suppose $X_n$ converges in distribution to $c$, so that $F_n(x) \rightarrow 1$ for $x>c$ and $F_n(x) \rightarrow 0$ for $x<c$. \pause Let $\epsilon>0$ be given. \pause

\begin{eqnarray*}
    P\{|X_n-c|<\epsilon\} & = & F_n(x+\epsilon)-F_n(x-\epsilon) \pause \mbox{ so} \\ \pause
    \lim_{n \rightarrow \infty}P\{|X_n-c|<\epsilon\} 
    & = & \lim_{n \rightarrow \infty}F_n(x+\epsilon) - 
    \lim_{n \rightarrow \infty}F_n(x-\epsilon)  \\ \pause
    & = & 1-0  \pause = 1 
\end{eqnarray*} \pause
And $X_n$ converges in distribution to $c$.
} % End size of whole slide.
\end{frame}

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\begin{frame}
\frametitle{Comment} 
%\framesubtitle{} 
\begin{itemize}
    \item Convergence in probability might seem redundant, because it's just convergence in distribution to a constant. \pause
    \item But that's only true when the convergence is to a constant. \pause
    \item Convergence in probability to a non-degenerate random variable \pause implies convergence in distribution. \pause
    \item But convergence in distribution 
    does not imply convergence in probability \pause when the convergence is to a non-degenerate variable.
\end{itemize}
\end{frame}

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\begin{frame}
\frametitle{Big Theorem about convergence in distribution}
\framesubtitle{Book calls it the ``Continuity Theorem"} \pause
Let the random variables $X_1, X_2 \ldots$ have cumulative distribution functions $F_1(x), F_2(x) \ldots$ and moment-generating functions $M_1(t), M_2(t) \ldots$. \pause Let the random variable $X$ have cumulative distribution function $F(x)$ and moment-generating function $M(t)$. \pause If
\begin{displaymath}
    \lim_{n \rightarrow \infty} M_n(t) = M(t)
\end{displaymath} \pause
for all $t$ in an open interval containing $t=0$, \pause then $X_n$ converges in distribution to $X$. \pause \vspace{5mm}

The idea is that convergence of moment-generating functions implies convergence of distribution functions. 
\end{frame}

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\begin{frame}
\frametitle{Example: Poisson approximation to the binomial}
\framesubtitle{We did this before with probability mass functions and it was a challenge.}  \pause
Let $X_n$ be a binomial ($n,p_n$) random variable with $p_n=\frac{\lambda}{n}$, so that $n \rightarrow \infty$ and $p \rightarrow 0$ in such a way that the value of $n \, p_n=\lambda$ remains fixed. Find the limiting distribution of $X_n$. \pause \vspace{1mm}

Recalling that the MGF of a Poisson is $e^{\lambda(e^t-1)}$ and $\left(1 + \frac{x}{n}\right)^n \rightarrow  e^x$, \pause

\begin{eqnarray*}
    M_n(t) & = & (pe^t+1-p)^n \\ \pause
    & = & \left(\frac{\lambda}{n}e^t+1-\frac{\lambda}{n} \right)^n \\ \pause
    & = & \left(1+\frac{\lambda(e^t-1}{n} \right)^n \\ \pause
    & \rightarrow & e^{\lambda(e^t-1)} \\ \pause
\end{eqnarray*} 
MGF of Poisson($\lambda$).
\end{frame}

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\begin{frame}
\frametitle{The Central Limit Theorem} \pause
%\framesubtitle{} 
Let $X_1, \ldots, X_n$ be independent random variables from a distribution with expected value $\mu$ and variance $\sigma^2$. \pause Then 

\begin{displaymath}
    Z_n = \frac{\sqrt{n}(\overline{X}_n-\mu)}{\sigma}
\end{displaymath} \pause
converges in distribution to $Z \sim$ Normal(0,1). \pause \vspace{5mm}

In practice, $Z_n$ is often treated as standard normal for $n>25$\pause, although the $n$ required for an accurate approximation really depends on the distribution.
\end{frame}


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\begin{frame}
\frametitle{Sometimes we say the distribution of the sample mean is approximately normal, or ``asymptotically" normal.} \pause
%\framesubtitle{} 
  \begin{itemize}
    \item This is justified by the Central Limit Theorem. \pause
    \item But it does \emph{not} mean that $\overline{X}_n$ converges in distribution to a normal random variable. \pause
    \item The Law of Large Numbers says that $\overline{X}_n$ converges in probability to a constant, $\mu$. \pause
    \item So $\overline{X}_n$ converges to $\mu$ in distribution as well. \pause
    \item That is, $\overline{X}_n$ converges in distribution to a degenerate random variable with all its probability at $\mu$.
  \end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Why would we say that for large $n$, the sample mean is approximately $N(\mu,\frac{\sigma^2}{n})$?} 

\pause

\vspace{5mm}

Have 
$Z_n = \frac{\sqrt{n}(\overline{X}_n-\mu)}{\sigma}$ \pause converging to
$ Z \sim N(0,1)$. 

\pause

{\footnotesize
\begin{eqnarray*}
    Pr\{\overline{X}_n \leq x\}  \pause
    & = & 
    Pr\left\{ \frac{\sqrt{n}(\overline{X}_n-\mu)}{\sigma} \leq
    \frac{\sqrt{n}(x-\mu)}{\sigma}\right\}    \\ \pause 
    & = & 
    Pr\left\{ Z_n \leq \frac{\sqrt{n}(x-\mu)}{\sigma}\right\}  \pause 
    \approx   \Phi\left( \frac{\sqrt{n}(x-\mu)}{\sigma} \right)
\end{eqnarray*} }

\pause

Suppose $Y$ is \emph{exactly} $N(\mu,\frac{\sigma^2}{n})$: 

\pause

{\footnotesize
\begin{eqnarray*}
    Pr\{Y \leq x\}  \pause
    & = & 
    Pr\left\{ \frac{\sqrt{n}(Y-\mu)}{\sigma} \leq
    \frac{x-\mu}{\sigma/\sqrt{n}}\right\}    \\ \pause 
    & = & 
    Pr\left\{ Z_n \leq \frac{\sqrt{n}(x-\mu)}{\sigma}\right\}  \pause 
    =  \Phi\left( \frac{\sqrt{n}(x-\mu)}{\sigma} \right)
\end{eqnarray*} 
} % End size
\end{frame}


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\begin{frame}
\frametitle{Copyright Information}

This slide show was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Statistical Sciences, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website:

\vspace{5mm}
\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f18} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f18}}
\end{frame}
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\end{document}


and $Var(Y) = \sigma^2$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{} \pause
%\framesubtitle{} 
\begin{itemize}
    \item  \pause
    \item  \pause
    \item 
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\includegraphics[width=2in]{BivariateNormal}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{}
%\framesubtitle{} 
{\LARGE
\begin{eqnarray*}
    m_1 & = & a + b \\
    m_2 & = & c + d 
\end{eqnarray*} 
} % End size
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{} \pause
%\framesubtitle{} 
\begin{itemize}
    \item  \pause
    \item  \pause
    \item 
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

# R code for plots of normal MGFs
tt = seq(from=-1,to=1,by=0.05)
mu = 0; sigsq = 1
zero = exp(mu*tt + 0.5*sigsq*tt^2)
mu = 1; one = exp(mu*tt + 0.5*sigsq*tt^2)
mu = -1; minusone = exp(mu*tt + 0.5*sigsq*tt^2)
x = c(tt,tt,tt); y = c(zero,one,minusone)
plot(x,y,pch=' ',xlab='t',ylab = 'M(t)')
lines(tt,zero,lty=1)
lines(tt,one,lty=2)
lines(tt,minusone,lty=3)
title("Fingerprints of the normal distribution")
# Legend
x1 <- c(-0.4,0) ; y1 <- c(4,4) ; lines(x1,y1,lty=1)
text(0.25,4,expression(paste(mu," =  0, ",sigma^2," = 1")))
x2 <- c(-0.4,0) ; y2 <- c(3.75,3.75) ; lines(x2,y2,lty=2)
text(0.25,3.75,expression(paste(mu," =  1, ",sigma^2," = 1")))
x3 <- c(-0.4,0) ; y3 <- c(3.5,3.5) ; lines(x3,y3,lty=3)
text(0.25,3.5,expression(paste(mu," = -1, ",sigma^2," = 1")))


# R code for plots of chi-squared MGFs
tt = seq(from=-0.25,to=0.25,by=0.005)
nu = 1; one = (1-2*tt)^(-nu/2)
nu = 2; two = (1-2*tt)^(-nu/2)
nu = 3; three = (1-2*tt)^(-nu/2)
x = c(tt,tt,tt); y = c(one,two,three)
plot(x,y,pch=' ',xlab='t',ylab = 'M(t)')
lines(tt,one,lty=1)
lines(tt,two,lty=2)
lines(tt,three,lty=3)
title("Fingerprints of the chi-squared distribution")
# Legend
x1 <- c(-0.2,-0.1) ; y1 <- c(2.5,2.5) ; lines(x1,y1,lty=1)
text(-0.05,2.5,expression(paste(nu," =  1")))
x2 <- c(-0.2,-0.1) ; y2 <- c(2.3,2.3) ; lines(x2,y2,lty=2)
text(-0.05,2.3,expression(paste(nu," =  2")))
x3 <- c(-0.2,-0.1) ; y3 <- c(2.1,2.1) ; lines(x3,y3,lty=3)
text(-0.05,2.1,expression(paste(nu," =  3")))