%% This document created by Scientific Notebook (R) Version 3.0


\documentclass[12pt,thmsa]{article}
\usepackage{amssymb}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{makeidx}
\usepackage{sw20jart}

%TCIDATA{TCIstyle=article/art4.lat,jart,sw20jart}

%TCIDATA{<META NAME="ViewSettings" CONTENT="31">}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="Title" CONTENT="Families of Discrete Distributions">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 16:51:51 1997}
%TCIDATA{Language=American English}
%TCIDATA{CSTFile=MathHelp.cst}
%TCIDATA{<LINK REL="begin" HREF="DM11-1.tex">}
%TCIDATA{<LINK REL="contents" HREF="DM0-0.tex">}
%TCIDATA{<LINK REL="end" HREF="DM11-9.tex">}
%TCIDATA{<LINK REL="next" HREF="DM11-7.tex">}
%TCIDATA{<LINK REL="parent" HREF="DM11.tex">}
%TCIDATA{<LINK REL="previous" HREF="DM11-5.tex">}
%TCIDATA{<LINK REL="top" HREF="DM0-0.tex">}

\input{tcilatex}
\begin{document}


\section{Families of Discrete Distributions\label{Discrete distributions}}

Several of the standard distributions are functions of a discrete variable,
usually the integers. They are commonly plotted with bar graphs or broken
line (polygonal) graphs.

\subsection{\ Binomial}

The \textsl{binomial distribution functions} are functions of a nonnegative
integer $x$, 
\[
\func{BinomialDist}(x;n,p)=\sum_{k=0}^{x}\binom{n}{k}p^{k}q^{n-k} 
\]
with Bernoulli trial parameter (or sample size) a positive integer $n$,
Bernoulli probability parameter a real number $p$ with $0<p<1$ and $q=1-p$.
(To enter 
\index{Binomial coefficients@Binomial coefficients}\label%
{Binomial coefficients}\textsl{binomial coefficients,} $%
\binom{n}{k}=\frac{n!}{k!\left( n-k\right) !}$, click the binomial fraction 
\FRAME{itbpF}{0.2992in}{0.2992in}{0.0692in}{}{}{binomial.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid-file "F";width 0.2992in;height 0.2992in;depth
0.0692in;original-width 17.8125pt;original-height 16.9375pt;cropleft
"-0.038101";croptop "1.0143161";cropright "0.974699";cropbottom
"0.0739161";filename 'BINOMIAL.WMF';file-properties "XNPEU";}}and choose 
\textsf{None} for line.) The corresponding \textsl{binomial probability
density function} is 
\[
\func{BinomialDen}(x;n,p)=\binom{n}{x}p^{x}q^{n-x} 
\]
for the same conditions on $x$, $n$, and $p$. The mean for this distribution
is $np,$ and the variance is $npq$. 
\index{Binomial distribution@Binomial distribution}\label%
{Binomial distribution}\medskip

Binomial distribution tables found in statistics books give selected values
of either the binomial probability density function $%
\func{BinomialDen}(x;n,p)$ or the cumulative distribution function $\func{%
BinomialDist}(x;n,p)$.

The binomial density $\func{BinomialDen}(x;n,p)$ gives the probability of $x$
successes in $n$ independent Bernoulli trials, when the probability of
success at each trial is $p$. It is by far the most common discrete
distribution, since people deal with many experiments in which a dichotomous
classification of the result is of primary interest. The name \textsl{%
binomial distribution} comes from the fact that the coefficients $\binom{n}{k%
}=\frac{n!}{k!\left( n-k\right) !}$ are commonly called binomial
coefficients.\medskip

\begin{example}
Compute the probability that, in $100$ tosses of a coin with $\Pr $(heads) $%
=.55$, no more than $54$ heads turn up, assuming a binomial distribution.

$\Pr (X\leq 54)=\func{BinomialDist}(54;100,.55)=.45846$
\end{example}

The binomial distribution function with parameters $n$ and $p$ can be
approximated by the normal distribution with mean $np$ and variance $%
np\left( 1-p\right) $; that is, 
\[
\func{BinomialDist}(x;np,np(1-p))\approx \func{NormalDist}(x;n,p) 
\]
Such approximations are reasonably good if $np$ and $n\left( 1-p\right) $
are greater than $5$. For example, to find an approximate solution to the
preceding problem using a normal distribution, use

\begin{eqnarray*}
\Pr (X &\leq &54)=\func{BinomialDist}(54;100,.55)=.45846 \\
&\approx &\func{NormalDist}(54;55.0,24.75)=.48389
\end{eqnarray*}

The following plot shows the graph of $\func{NormalDist}(x;55.0,24.75)$ and
a polygonal plot of $\func{BinomialDist}(x;100,.55)$ for $0\leq x\leq 100$.

\FRAME{dtbpFU}{3in}{2.0003in}{0pt}{\Qcb{Normal and Binomial distributions}}{%
}{}{\special{language "Scientific Word";type "MAPLEPLOT";width 3in;height
2.0003in;depth 0pt;display "PICT";plot-snapshots TRUE;function
\TEXUX{$\left( 0,0,30,3.83\times 10^{-7},35,4.45\times 10^{-5},40,1.82\times
10^{-3},45,2.84\times 10^{-2},50,.183,65,.98,70,.99,80,1.0,90,1,100,1\right)
$};linecolor "blue";linestyle 1;linethickness 1;pointstyle "point";function
\TEXUX{$2.\,0202\times 10^{-2}\frac{\sqrt{2}}{\sqrt{\pi }}\int_{-\infty
}^{x}\exp \left( -8.\,1624\times 10^{-4}\left( u-55\right) ^{2}\right)
\,du$};linecolor "red";linestyle 1;linethickness 1;pointstyle "point";xmin
"0";xmax "100";xviewmin "-2";xviewmax "102";yviewmin "-0.02";yviewmax
"1.02";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle
"normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}

\ 

\subsection{\ Poisson 
\index{Poisson distribution@Poisson distribution}\label{Poisson distribution}
}

The \textsl{Poisson cumulative distribution function} is a discrete function
defined for non-negative integers $x$. The Poisson distribution with mean $%
\mu >0$ is defined by the summation 
\[
\func{PoissonDist}\left( x;\mu \right) =\sum_{k=0}^{x}\frac{\mu ^{k}e^{-\mu }%
}{k!} 
\]
The \textsl{Poisson probability density function} is 
\[
\limfunc{PoissonDen}\left( k;\mu \right) =\frac{\mu ^{k}e^{-\mu }}{k!} 
\]
for nonnegative integers $k$ and real numbers $\mu >0$. A Poisson
distribution table lists selected values of the Poisson probability density
function $\limfunc{PoissonDen}\left( k;\mu \right) $.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate Numerically}
\end{quote}

\begin{quotation}
$\limfunc{PoissonDen}\left( 2;3\right) =.22404$

$\limfunc{PoissonDen}\left( 5;0.3\right) =1.5002\times 10^{-5}$

$\limfunc{PoissonDen}\left( 10;4\right) =5.2925\times 10^{-3}$
\end{quotation}

The Poisson distribution can be used to approximate the binomial
distribution when the probability is small and $n$ is large; that is, 
\[
\func{PoissonDist}(k;\mu )\approx \func{BinomialDist}\left( k;\mu ,\mu
\left( 1-p\right) \right) 
\]
where $\mu =np$. This distribution has been used as a model for a variety of
random phenomena of practical importance.\medskip

\subsection{\ Hypergeometric 
\index{Hypergeometric distribution@Hypergeometric distribution}\label%
{Hypergeometric distribution}}

Suppose that, from a population of $M$ elements, of which $x$ possess a
certain attribute, you draw a sample of $n$ items without replacement. The
number of items that possess the certain attribute in such a sample is a
hypergeometric variate. The \textsl{hypergeometric cumulative distribution
function} is a discrete function defined for nonnegative integers $x$. The
hypergeometric distribution with $M$ elements in the population, $K$
successes in the population, and sample size $n$ is defined by the following
summation of quotients of binomial coefficients for $0\leq x\leq n.$ 
\[
\func{HypergeomDist}\left( x;M,K,n\right) =\sum_{k=0}^{x}\frac{\binom{K}{k}%
\binom{M-K}{n-k}}{\binom{M}{n}} 
\]
For $x<0,$ the distribution function is $0,$ and for $x\geq n,$ the function
is $1$. The \textsl{hypergeometric probability density function} is 
\[
\limfunc{HypergeomDen}\left( k;M,K,n\right) =\frac{\binom{K}{k}\binom{M-K}{%
n-k}}{\binom{M}{n}} 
\]
for integers $k$ , $K$ , $n$, and $M$ satisfying $0\leq k\leq n$, $0\leq
K\leq M,$ and $0<n\leq M$.

\begin{example}
What is the probability of at most five successes when you draw a sample of
10 from a population of 100, of which 30 members are identified as
successes? The probability of exactly $x$ successes is given by the function 
$\limfunc{HypergeomDen}\left( x;100,30,10\right) $. Thus, the probability of
at most five successes is the sum of exactly 0, 1, 2, 3, 4, and 5 successes,
or $\func{HypergeomDist}\left( 5;100,30,10\right) =.96123$
\end{example}

The hypergeometric distribution is the model for sampling without
replacement. The hypergeometric distribution can be approximated by the
binomial distribution when the sample size is relatively small.

These plots (created as polygonal plots) depict $\func{HypergeomDen}\left(
x;100,30,10\right) $ $\ $and $\func{HypergeomDist}\left( x;100,30,10\right) $
for $0\leq x\leq 10$.

\begin{center}
\FRAME{itbpFU}{2.1491in}{1.7495in}{0in}{\Qcb{Hypergeometric density function}%
}{}{Plot }{\special{language "Scientific Word";type "MAPLEPLOT";width
2.1491in;height 1.7495in;depth 0in;display "PICT";function \TEXUX{$\left(
0,0,1,0,1,.11,1,0,2,0,2,.24,2,0,3,0,3,.28,3,0,4,0,4,.21,4,0,5,0,5,.1,5,0,6,0,6,3.1\times 10^{-2},6,0,7,0,7,6.4\times 10^{-3},7,0,8,0,8,8.2\times 10^{-4},8,0,9,0,9,5.8\times 10^{-5},9,0,10,0,10,1.7\times 10^{-6}\right) $};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";xmin "0";xmax "10";xviewmin "0";xviewmax "10";yviewmin "-0.0056";yviewmax "0.4";viewset"XY";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{x};var1name \TEXUX{$x$};}%
} \FRAME{itbpFU}{2.1491in}{1.7495in}{0in}{\Qcb{Hypergeometric distribution
function}}{}{Plot }{\special{language "Scientific Word";type
"MAPLEPLOT";width 2.1491in;height 1.7495in;depth 0in;display "PICT";function
\TEXUX{$\left(
0,0,1,0,1,.14,1,0,2,0,2,.37,2,0,3,0,3,.65,3,0,4,0,4,.86,4,0,5,0,5,.96,5,0,6,0,6,.99,6,0,7,0,7,1,7,0,8,0,8,1,8,0,9,0,9,1,9,0,10,0,10,1\right) $};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";xmin "0";xmax "10";xviewmin "-0.2";xviewmax "10.2";yviewmin "-0.02";yviewmax "1.02";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{x};var1name \TEXUX{$x$};}%
}
\end{center}

\strut

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Menu Items, Lists, and Matrices}{}{}{DM11-1.tex#Statistics}

\item  \hyperref{Measures of Central Tendency}{}{}{DM11-2.tex#Statistics,
mean}

\item  \hyperref{Measures of Dispersion}{}{}{DM11-3.tex#Mean deviation}

\item  \hyperref{Distributions and Densities}{}{}{DM11-4.tex#Distribution}

\item  \hyperref{Families of Continuous Distributions}{}{}{%
DM11-5.tex#Continuous distributions}

\item  \hyperref{Random Numbers}{}{}{DM11-7.tex#Statistics, random numbers}

\item  \hyperref{Curve Fitting}{}{}{DM11-8.tex#Curve fitting}

\item  \hyperref{Exercises and Solutions}{}{}{DM11-9.tex#Exercises}
\end{itemize}

\rule{0.67in}{0.01in}

\hyperref{\FRAME{itbpF}{0.5275in}{0.3399in}{0.0519in}{}{}{index.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid-file
"F";width 0.5275in;height 0.3399in;depth 0.0519in;original-width
36.875pt;original-height 22.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'index.wmf';file-properties "XNPEU";}}}{}{}{%
DMIndex.tex#Math Index}\quad {\small Index entries: \hyperref{Distribution}{%
}{}{DMIndex.tex#Distribution}, \hyperref{Matrices}{}{}{DMIndex.tex#Matrices}%
, \hyperref{Statistics}{}{}{DMIndex.tex#Statistics}}

\end{document}
