% This is a simple LaTex sample document that gives a submission format
%   for IEEE PAMI-TC conference submissions.  Use at your own risk.

% Make two column format for LaTex 2e.


\documentclass[11pt,twocolumn]{article} %
\usepackage[dvips]{graphicx,graphics,color}
\usepackage{times,amsmath,amsfonts}

\setlength{\textheight}{10in} \setlength{\textwidth}{6.875in}
\setlength{\columnsep}{0.3125in} \setlength{\topmargin}{-0.5in}
\setlength{\headheight}{0in} \setlength{\headsep}{0in}
\setlength{\parindent}{1pc}
\setlength{\oddsidemargin}{-.2in}  % Centers text.
\setlength{\evensidemargin}{-.2in}

% Add the period after section numbers.  Adjust spacing.
\newcommand{\Section}[1]{\vspace{-8pt}\section{\hskip -1em.~~#1}\vspace{-3pt}}
\newcommand{\SubSection}[1]{\vspace{-3pt}\subsection{\hskip -1em.~~#1}
        \vspace{-3pt}}
\newcommand{\bqn}{\begin{eqnarray}}
\newcommand{\eqn}{\end{eqnarray}}
\newcommand{\bq}{\begin{eqnarray*}}
\newcommand{\eq}{\end{eqnarray*}}
\begin{document}

% Don't want date printed
\date{February 10, 2005}

% Make title bold and 14 pt font (Latex default is non-bold, 16pt)
\title{Stat 324: Lecture 08\\
Continuous distributions}

% For single author (just remove % characters)
\author{Moo K. Chung\\
mchung@stat.wisc.edu}

% For two authors (default example)
\maketitle \thispagestyle{empty}
\begin{enumerate}


\item  The {\em cumulative distribution function} $F$
of a continuous random variable $X$ is defined as
$$F(x) \leq P(X \leq x) = \int_{-\infty}^x f(x) \;dx.$$
From the fundamental theorem of calculus, the density function $f$
can be obtained as $f(x)=F'(x)$.
\item For given $p$, the $100p$-th {\em percentile} of the distribution of $X$ is a number
$\eta(p)$ such that
$$p = \int_{-\infty}^{\eta(p)} f(x) \;dx.$$
\begin{verbatim}
> x=0:40/10
>plot(x,dunif(x,1,3),type="s")
> qunif(0.3,1,3)
[1] 1.6
\end{verbatim}
\begin{figure}[h]
\centering
\includegraphics[width=50mm]{324.08.01.ps}
\caption{Density function of $Unif(2,3)$.}\label{fig:hist}
\end{figure}

\item The {\em expected value} of a continuous random variable X
with pdf $f(x)$ is $$\mathbb{E} X = \int_{-\infty}^{\infty} xf(x)
\;dx.$$
$$\mathbb{E} h(X) = \int_{-\infty}^{\infty} h(x)f(x)
\;dx.$$

\item A continuous random variable $X$ is a normal distribution with
parameters $\mu$ and $\sigma^2$ if $$f(x) =
\frac{1}{\sqrt{2\pi}\sigma}e^{-(x-\mu)^2/2\sigma^2}.$$ We will
denote this distribution $$X \sim N(\mu,\sigma^2).$$ When $Z \sim
N(0,1)$, it is a standard normal distribution. The cdf of $Z$,
$$\Phi(z) = \int_{-\infty}^z
\frac{1}{\sqrt{2\pi}}e^{-x^2/2\sigma^2}\;dz.$$
\begin{verbatim}
>x=-100:100/10
>plot(x,dnorm(x,0,1),type="l")
\end{verbatim}
\begin{figure}
\centering
\includegraphics[width=50mm]{324.08.02.ps}
\caption{Density function of standard normal
$N(0,1)$}\label{fig:hist}
\end{figure}

\item The $100(1-\alpha)$-th percentile of $N(0,1)$ will be denoted by
 $z_{\alpha}$.
\begin{verbatim}
> alpha<-c(0,0.01,0.025,0.05,0.1)
> alpha
[1] 0.000 0.010 0.025 0.050 0.100
> qnorm(alpha,0,1)
[1]      -Inf -2.33 -1.96 -1.64 -1.28
> qnorm(1-alpha,0,1)
[1]      Inf 2.33 1.96 1.64 1.28
\end{verbatim}
\end{enumerate}
 {\bf Assigned Problems} Exercises 4.18., 4.32. Read section 4.3. The first midterm will cover
 upto Lecture 09.


\end{document}