% 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}
%\usepackage{pancyhdr}
\setlength{\textheight}{10.5in} \setlength{\textwidth}{7in}
\setlength{\columnsep}{0.3in} \setlength{\topmargin}{-0.5in}
\setlength{\headheight}{-0.2in} \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
% Make title bold and 14 pt font (Latex default is non-bold, 16pt)

\title{Stat 311-Lecture 07 Variance}
% For single author (just remove % characters)
%\author{\Large Probability}
\vspace{-2cm}
\date{ }
% For two authors (default example)
\maketitle \thispagestyle{empty}

Sept 28, 2005


\begin{enumerate}


\item The variance of random variable is defined as $\mathbb{V} X = \mathbb{E} (X-\mathbb{E} X)^2$. 

\item Theorem. $\mathbb{V} X = \mathbb{E} X^2 - (\mathbb{E} X)^2.$\\
$\mathbb{V} (aX+b) = a^2 \mathbb{V} X$.



\item For $X \sim
Binomial(n,p)$, $\mathbb{E}X=np$. $$\mathbb{E}X(X-1)=n(n-1)p^2, \mathbb{V} X = np(1-p).$$


\item $X$ is a Poisson random variable with parameter $\lambda$ if $p(x) = e^{-\lambda}\frac{\lambda^x}{x!}, x=0,1,2,\cdots$. Note that
$$e^{\lambda} = 1 + \lambda + \frac{\lambda^2}{2!}+ \cdots.$$

$\mathbb{E} X =\mathbb{V} X = \lambda$.

\item Problem. Suppose the probability of containing at least one typographical error in a page of a book is 0.005. Assuming typographical errors occur independently from page to page, what is the probability that a 400 page novel will contain exactly one page with errors?

{\em Solution.} If $X$ is the number of pages containing at least one error. It is distributed as $Binomial(400,0.005)$. Then $P(X=1) = {400 \choose 1}0.005\cdot0.995^{399}=0.2707$. Your calculator may have hard time computing this probability. So we approximate this with $Poission(2)$. $P(X=1) \approx e^{-2}\cdot2 = 0.2707$.

\item Iterative procedure for computing Poisson probability function. 
$$p(x+1) = \frac{\lambda}{x+1}p(x).$$
Continuing the previous problem, what is the probability that a 400 page novel will contain at most 3 pages with errors? 
{\em Solution.} We need to compute $P(X \leq 3) = p(0) + p(1) + p(2) + p(3)$. We compute $p(0)$ first. $p(0) = e^{-2} = 0.1353$. $p(1) = 2p(0)=0.2707$, $p(2)=p(1)=0.2707$, $p(3) = \frac{2}{3}p(2)=0.180$. 

\item Problem. The number of cars arriving at a vehicle equipment inspection station follows Possion with parameter 10. 50$\%$ of arriving cars will have broken equipments. What is the probability exactly 10 cars will arrive during the hour and all have no broken equipment?\\
{\em Solution.} Let $X$ be the number of arriving cars and $Y$ be the number of cars with broken equipments. $P(X=10, Y=0) = P(Y=0|X=10)P(X=10) = 0.5^{10} e^{-10}10^{10}/10!$.

What is the probability that $c \geq 10$ cars arrive, of which 10 have no broken equipment?\\
{\em Solution.} 
\bq P(X=c, Y=10) &=& P(Y=10|X=c)P(X=c)\\
&=& {c \choose 10}0.5^c e^{-10}^c/c!\\
&=&e^{-10}\frac{5^c}{10!(c-10)!}\eq
What is the probability that 10 cars with no broken equipment arrives during the next hour? (This means more than 10 cars will arrive but at least 10 of them will have no broken equipment.)
$$\sum_{c=10}^{\infty}  e^{-10}\frac{5^c}{10!(c-10)!}=e^{-5}5^{10}/10!$$
Probabily mass function for $Possion(5)$.
\end{enumerate}




\end{document}