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%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 16:51:19 1997}
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\input{tcilatex}
\begin{document}


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

The relationship of the various distribution, inverse distribution, and
density functions to the entries in standard statistical tables is explained
below for each of the families of distributions available in \textsl{%
Scientific Notebook}.

\subsection{\ Normal\label{Normal distribution}}

The \textsl{normal cumulative distribution function} is 
\index{Normal distribution@Normal distribution}defined for all real numbers $%
\mu $ and for positive $\sigma $ by the integral 
\[
\limfunc{NormalDist}\left( x;\mu ,\sigma \right) =\frac{1}{\sigma \sqrt{2\pi 
}}\int_{-\infty }^{x}e^{-\frac{\left( u-\mu \right) ^{2}}{2\sigma ^{2}}}du 
\]
of the \textsl{normal probability density function} 
\[
\limfunc{NormalDen}\left( u;\mu ,\sigma \right) =\frac{1}{\sigma \sqrt{2\pi }%
}e^{-\frac{\left( u-\mu \right) ^{2}}{2\sigma ^{2}}} 
\]
The inverse of the normal cumulative distribution function, $\limfunc{%
NormalInv}$, is also available. All three of these functions can be typed in
mathematics, and they will automatically turn gray as you type the final
letter.

The parameters $\mu $ and $\sigma $ are optional parameters for mean and
standard deviation, with the default values $0$ and $1$ defining the \textsl{%
standard normal distribution } 
\[
\limfunc{NormalDist}\left( x\right) =\frac{1}{\sqrt{2\pi }}\int_{-\infty
}^{x}e^{-\frac{u^{2}}{2}}du 
\]
A normal distribution table, as found in the back of a typical statistics
book, lists some values of the standard normal cumulative distribution
function. Certain versions of the table list the values $1-\limfunc{%
NormalDist}\left( x\right) $.

Note that the function $\limfunc{NormalDist}$ can be evaluated as a function
of one variable (with default parameters $(0,1)$) or as a function of one
variable and two parameters.

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

\begin{itemize}
\item  $\limfunc{NormalDist}(2.44)=\allowbreak .\,\allowbreak 992\,66$

\item  $\limfunc{NormalDist}(2.44;0,1)=\,.99266$

\item  $\limfunc{NormalDist}(2.44;1,2)=\,.76424$

\item  $\func{NormalDen}(2.44;1,2)=\allowbreak .\,\allowbreak 153\,93$
\end{itemize}

Graphs of the normal density functions are the familiar bell-shaped curves.
The following plots show the density functions $\func{NormalDen}\left( x;\mu
,\sigma \right) $ and distribution functions $\limfunc{NormalDist}\left(
x;\mu ,\sigma \right) $ for the parameters $\left( \mu ,\sigma \right)
=\left( 0,1\right) ,\left( 0,5\right) ,\left( 0,.5\right) ,\left( 1,1\right) 
$.

\begin{center}
\FRAME{itbpFU}{2.1577in}{1.7573in}{0in}{\Qcb{Normal density functions}}{}{}{%
\special{language "Scientific Word";type "MAPLEPLOT";width 2.1577in;height
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\frac{1}{2}\frac{\sqrt{2}}{\sqrt{\pi }}e^{-\frac{1}{2}x^{2}}$};linecolor
"blue";linestyle 1;linethickness 1;pointstyle "point";function
\TEXUX{$\allowbreak \frac{1}{10}\frac{\sqrt{2}}{\sqrt{\pi
}}e^{-\frac{1}{50}x^{2}}$};linecolor "green";linestyle 1;linethickness
1;pointstyle "point";function \TEXUX{$\allowbreak
1.0\frac{\sqrt{2}}{\sqrt{\pi }}\exp \left( -2.0x^{2}\right) $};linecolor
"magenta";linestyle 1;linethickness 1;pointstyle "point";function
\TEXUX{$\allowbreak \frac{1}{2}\frac{\sqrt{2}}{\sqrt{\pi
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"normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}$\allowbreak $\FRAME{itbpFU}{%
2.1577in}{1.7573in}{0in}{\Qcb{Normal distribution functions}}{}{}{%
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1.0\frac{\sqrt{2}}{\sqrt{\pi }}\int_{-\infty }^{x}\exp \left(
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"5";xviewmin "-5";xviewmax "5";yviewmin "-0.02";yviewmax
"1.02";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle
"normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}
\end{center}

\ 

\subsection{\ Student's t\label{Student's t distribution}\label%
{Gamma function (1)}}

The \textsl{gamma function} $\Gamma (t)$ 
\index{Gamma function@Gamma function}that appears 
\index{Student's t@Student's t} in the definition of the Student's t
distribution (and the gamma distribution) is the continuous function $\Gamma
\left( t\right) =\int_{0}^{\infty }e^{-x}x^{t-1}dx$ defined for positive
real numbers $t$. The gamma function satisfies 
\[
\Gamma (1)=\,1%
\text{ and }\Gamma (t+1)=t\Gamma (t) 
\]
and for positive integers $k$, it is the familiar factorial function 
\index{Factorial@Factorial} 
\[
\Gamma (k)=\left( k-1\right) ! 
\]
The 
\hyperref{gamma function}{}{}{DM6-4.tex#Gamma function} is active in \textsl{%
Scientific Notebook}. For example, place the insertion point in the
expression $\Gamma (5)$ and choose \textsf{Evaluate} to get $\Gamma
(5)=\,24. $ Note that $24=4\times 3\times 2\times 1$.

The \textsl{Student's t cumulative distribution function} $\limfunc{TDist}%
(x;v)$ is defined by the integral 
\[
\limfunc{TDist}(x;v)=\frac{\Gamma (\frac{v+1}{2})}{\Gamma (\frac{v}{2})\sqrt{%
\pi v}}\int_{-\infty }^{x}\left( 1+\frac{1}{v}u^{2}\right) ^{-\frac{v+1}{2}%
}du 
\]
of the density function 
\[
\limfunc{TDen}(u;v)=\frac{\Gamma (\frac{v+1}{2})}{\Gamma (\frac{v}{2})\sqrt{%
\pi v}}\left( 1+\frac{1}{v}u^{2}\right) ^{-\frac{v+1}{2}} 
\]
with shape parameter $v$, called degrees of freedom, that ranges over the
positive integers. The variance for a Student's t distribution is $\frac{v}{%
v-2}$, provided $v>2$.

The function $\limfunc{TInv}(p;v)$ is the value of $x$ for which the
integral has the value $p$, as demonstrated here: 
\[
\begin{array}{lll}
\limfunc{TDist}(63.66;1)=\,.995 &  & \limfunc{TDist}(-.97847;3)=\,.2 \\ 
\limfunc{TInv}(.995;1)=\,63.66 &  & \limfunc{TInv}(.2;3)=\,-.97847
\end{array}
\]

The following plots display the density and distribution functions $\func{%
TDen}\left( x;v\right) $ and $\func{TDist}\left( x;v\right) $ for the
parameters $v=1$ and $v=15\ $ with $-5\leq x\leq 5$.

\begin{center}
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functions}}{}{}{\special{language "Scientific Word";type "MAPLEPLOT";width
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4;numpoints 49;axesstyle "normal";xis \TEXUX{u};var1name \TEXUX{$x$};}} 
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45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis
\TEXUX{x};var1name \TEXUX{$x$};}}
\end{center}

Note that the Student's t density functions resemble the standard normal
density function in shape, although these curves are a bit flatter at the
center. It is not difficult to show, using \textsl{Scientific Notebook} and
the definitions of the two density functions, that $\lim_{v\rightarrow
\infty }\limfunc{TDen}(u;v)=$ $\limfunc{NormalDen}(u),$ the density function
for the standard normal distribution.

Student's t distribution tables list values of the inverse distribution
function corresponding to probabilities (values of the distribution
function) and degrees of freedom. For values of $v$ above $30,$ the normal
distribution is such a close approximation for the Student's t distribution
that tables usually provide values only up to $v=30$.\medskip

\begin{example}
Assuming a Student's t distribution with $5$ degrees of freedom, determine a
value $c$ such that $\ \limfunc{Pr}(-c<T<c)=.90$, where $\limfunc{Pr}$
denotes probability. Now 
\begin{eqnarray*}
\Pr (-c &<&T<c)=\Pr \left( T\leq c\right) -\Pr \left( T\leq -c\right) \\
\qquad &=&\limfunc{TDist}\left( c;5\right) -\limfunc{TDist}\left( -c;5\right)
\end{eqnarray*}
So, you need to solve 
\[
\limfunc{TDist}\left( c;5\right) -\limfunc{TDist}\left( -c;5\right) =.90 
\]
The Student's t distribution satisfies 
\[
\limfunc{TDist}\left( c;5\right) +\limfunc{TDist}\left( -c;5\right) =1 
\]
So, the problem reduces to 
\begin{eqnarray*}
2\limfunc{TDist}\left( c;5\right) -1 &=&.90 \\
\limfunc{TDist}\left( c;5\right) &=&\frac{.90+1}{2}=.95
\end{eqnarray*}
The problem is solved by 
\[
\limfunc{TInv}(.95;5)=2.015 
\]
\medskip
\end{example}

\subsection{\ Chi-Square\label{Chi-Square distribution}}

The \textsl{chi-square cumulative distribution function} is 
\index{Chi-square distribution@Chi-square distribution}defined for
nonnegative $x$ and $\mu $ by the integral 
\[
\func{ChiSquareDist}(x;\mu )=\frac{1}{\Gamma (\frac{\mu }{2})2^{\frac{\mu }{2%
}}}\int_{0}^{x}u^{\frac{\mu }{2}-1}e^{-\frac{u}{2}}du 
\]
of the \textsl{chi-square probability density function } 
\[
\limfunc{ChiSquareDen}(u;\mu )=\frac{1}{\Gamma (\frac{\mu }{2})2^{\frac{\mu 
}{2}}}u^{\frac{\mu }{2}-1}e^{-\frac{u}{2}} 
\]
The indexing parameter $\mu >0$ is the mean of the distribution; it is
referred to as the degrees of freedom.

The following plots show distribution functions $\func{ChiSquareDist}(x;\mu
) $ and density functions $\limfunc{ChiSquareDen}(x;\mu )$ for $\mu
=1,5,10,15$ and $0\leq x\leq 25$.

\begin{center}
\bigskip \FRAME{itbpFU}{2.1577in}{1.7573in}{0in}{\Qcb{Chi-square density
functions}}{}{}{\special{language "Scientific Word";type "MAPLEPLOT";width
2.1577in;height 1.7573in;depth 0in;display "PICT";function
\TEXUX{$\frac{1}{2\sqrt{\pi
}}\frac{\sqrt{2}}{\sqrt{u}}e^{-\frac{1}{2}u}$};linecolor "blue";linestyle
1;linethickness 1;pointstyle "point";function \TEXUX{$\frac{1}{6\sqrt{\pi
}}\sqrt{2}\left( \sqrt{u}\right) ^{3}e^{-\frac{1}{2}u}$};linecolor
"green";linestyle 1;linethickness 1;pointstyle "point";function
\TEXUX{$\frac{1}{270270\sqrt{\pi }}\sqrt{2}\left( \sqrt{u}\right)
^{13}e^{-\frac{1}{2}u}$};linecolor "magenta";linestyle 1;linethickness
1;pointstyle "point";function
\TEXUX{$\frac{1}{768}u^{4}e^{-\frac{1}{2}u}$};linecolor "red";linestyle
1;linethickness 1;pointstyle "point";xmin "0";xmax "25";xviewmin
"0";xviewmax "25";yviewmin "0";yviewmax "0.2";viewset"XY";rangeset"X";phi
45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis
\TEXUX{u};var1name \TEXUX{$x$};}} \FRAME{itbpFU}{2.1577in}{1.7573in}{0in}{%
\Qcb{Chi-square distribution functions}}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 2.1577in;height 1.7573in;depth 0in;display
"PICT";function \TEXUX{$\frac{1}{2\sqrt{\pi
}}\sqrt{2}\int_{0}^{x}\frac{1}{\sqrt{u}}e^{-\frac{1}{2}u}\,du$};linecolor
"blue";linestyle 1;linethickness 1;pointstyle "point";function
\TEXUX{$\frac{1}{6\sqrt{\pi }}\sqrt{2}\int_{0}^{x}\left( \sqrt{u}\right)
^{3}e^{-\frac{1}{2}u}\,du$};linecolor "magenta";linestyle 1;linethickness
1;pointstyle "point";function
\TEXUX{$\frac{1}{768}\int_{0}^{x}u^{4}e^{-\frac{1}{2}u}\,du$};linecolor
"red";linestyle 1;linethickness 1;pointstyle "point";function
\TEXUX{$\frac{1}{270270\sqrt{\pi }}\sqrt{2}\int_{0}^{x}\left(
\sqrt{u}\right) ^{13}e^{-\frac{1}{2}u}\,du$};linecolor "green";linestyle
1;linethickness 1;pointstyle "point";xmin "0";xmax "25";xviewmin
"0";xviewmax "25";yviewmin "0";yviewmax "1";viewset"XY";rangeset"X";phi
45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis
\TEXUX{x};var1name \TEXUX{$x$};}}
\end{center}

The function, $\limfunc{ChiSquareInv}\left( t;\nu \right) \ $gives the value
of $x$ for which $\func{ChiSquareDist}(x;\nu )=t$. This relationship is
demonstrated in the following examples. 
\begin{eqnarray*}
\func{ChiSquareDist}(1.6103;5) &=&\,9.9999\times 10^{-2}\approx .1 \\
\limfunc{ChiSquareInv}(.1;5) &=&\,1.6103 \\
\func{ChiSquareDist}\left( 2.366;3\right) &=&\,.5 \\
\limfunc{ChiSquareInv}(.5;3) &=&\,2.366
\end{eqnarray*}

A chi-square distribution table shows values of $\nu $ down the left column
and values $u$ of $\func{ChiSquareDist}$ across the top row. The entry in
row $\nu $ and column $u$ is $\limfunc{ChiSquareInv}(u;\nu )$.\medskip

\subsection{\ F\label{F distribution}}

The \textsl{F cumulative distribution function} is 
\index{F distribution@F distribution}given by the integral 
\[
\limfunc{FDist}(x;n,m)=\frac{\Gamma (\frac{n+m}{2})}{\Gamma (\frac{n}{2}%
)\Gamma (\frac{m}{2})}\left( \frac{n}{m}\right) ^{\frac{n}{2}}\int_{0}^{x}u^{%
\frac{n-2}{2}}\left( 1+\frac{n}{m}u\right) ^{-\frac{n+m}{2}}du 
\]
of the probability density function 
\[
\limfunc{FDen}(u;n,m)=\frac{\Gamma (\frac{n+m}{2})}{\Gamma (\frac{n}{2}%
)\Gamma (\frac{m}{2})}\left( \frac{n}{m}\right) ^{\frac{n}{2}}u^{\frac{n-2}{2%
}}\left( 1+\frac{n}{m}u\right) ^{-\frac{n+m}{2}} 
\]
The variable $x$ can be any positive number, and $n$ and $m$ can be any
positive integers. The F distribution is used to determine the validity of
the assumption of identical standard deviations of two normal populations.
It is the distribution on which the analysis of variance procedure is based.

The inverse distribution function $\limfunc{FInv}(p;n,m)$ gives the value of 
$x$ for which the integral $\limfunc{FDist}(x;n,m)$ has the value $p$. These
function names automatically turn gray when they are entered in mathematics
mode. The relationship between these two functions is illustrated in the
following examples.

\begin{eqnarray*}
\limfunc{FDist}(.1;3,5) &=&\,\,4.3419\times 10^{-2} \\
\limfunc{FInv}(.043419;3,5) &=&\,.1 \\
\limfunc{FDist}(3.7797;2,5) &=&\,.9 \\
\limfunc{FInv}(.9;2,5) &=&\,3.7797
\end{eqnarray*}

Standard F distribution tables list some of the values of the inverse F
distribution function. Thus, for example, the $4.4$th percentile for the F
distribution having degrees of freedom $\left( 3,5\right) $ is $\func{FInv}%
(.044;3,5)=.1,$ and the $90$th percentile for the F distribution having
degrees of freedom $\left( 2,5\right) $ is $\func{FInv}(.90;2,5)=3.7797$.

The following plots show distribution functions $\limfunc{FDist}(x;n,m)$ and
density functions $\func{FDen}(x;n,m)$ for $\left( n,m\right) =\left(
1,1\right) ,\left( 2,5\right) ,\left( 3,15\right) $, and $0\leq x\leq 5$.

\begin{center}
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"blue";linestyle 1;linethickness 1;pointstyle "point";function
\TEXUX{$\frac{1}{\left( \sqrt{\left( 1+\frac{2}{5}u\right) }\right)
^{7}}$};linecolor "green";linestyle 1;linethickness 1;pointstyle
"point";function \TEXUX{$\frac{32768}{10725\pi
}\sqrt{5}\frac{\sqrt{u}}{\left( 1+\frac{1}{5}u\right) ^{9}}$};linecolor
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4;numpoints 49;axesstyle "normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}
\end{center}

\ 

\subsection{\ Exponential\label{Exponential distribution}}

The \textsl{exponential cumulative distribution function with parameter }$%
\mu $, or mean $\mu $,\textsl{\ }is defined 
\index{Exponential distribution@Exponential distribution}by the integral 
\[
\func{ExponentialDist}\left( x;\mu \right) =\frac{1}{\mu }\int_{0}^{x}e^{%
\frac{-u}{\mu }}du=\,1-e^{\frac{-x}{\mu }} 
\]
of the density function 
\[
\limfunc{ExponentialDen}(u;\mu )=\frac{1}{\mu }e^{\frac{-u}{\mu }} 
\]
for $x\geq 0,$ and is $0$ otherwise.

The \textsl{inverse exponential distribution function } 
\[
\limfunc{ExponentialInv}\left( \alpha ;\mu \right) =\mu \ln \frac{1}{%
1-\alpha } 
\]
is the value of $x$ for which the$\,$ integral has the value$\ \,\alpha $,
as illustrated by the following: 
\begin{eqnarray*}
\limfunc{ExponentialInv}\left( .73;.58\right) &=&.58\ln \frac{1}{1-.73}%
=.75941 \\
\func{ExponentialDist}\left( .75941;.58\right) &=&1-e^{\frac{-.75941}{.58}%
}=.73
\end{eqnarray*}

The following plots show distribution functions $\func{ExponentialDist}%
\left( x;\mu \right) $ and density functions $\func{ExponentialDen}(x;\mu )$%
, for the parameters $\mu =1,3,5\ $ and $-2\leq x\leq 4$.

\begin{center}
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}{}{\special{language "Scientific Word";type "MAPLEPLOT";width
2.1577in;height 1.7573in;depth 0in;display "PICT";function
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"point";function \TEXUX{$\frac{1}{3}e^{-\frac{1}{3}u}$};linecolor
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\TEXUX{$1-e^{-\frac{1}{3}x}$};linecolor "green";linestyle 1;linethickness
1;pointstyle "point";function \TEXUX{$1-e^{-\frac{1}{5}x}$};linecolor
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"1";viewset"XY";rangeset"X";phi 45;theta 45;plottype 4;numpoints
49;axesstyle "normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}

\strut
\end{center}

\subsection{\ Weibull\label{Weibull distribution}}

The \textsl{Weibull distribution} with 
\index{Weibull distribution@Weibull distribution}scale parameter\textsl{\ }$%
b>0$ and shape parameter $a>0$ is defined by the integral 
\[
\func{WeibullDist}\left( x;a,b\right)
=ab^{-a}\int_{0}^{x}u^{a-1}e^{-u^{a}b^{-a}}du=1-e^{-x^{a}b^{-a}} 
\]
of the density function 
\[
\limfunc{WeibullDen}(u;a,b)=ab^{-a}u^{a-1}e^{-u^{a}b^{-a}} 
\]
for $x\geq 0$, and is $0$ otherwise.

The \textsl{inverse Weibull distribution function } 
\[
\limfunc{WeibullInv}\left( \alpha ;a,b\right) =b\left( \ln \frac{1}{1-\alpha 
}\right) ^{\frac{1}{a}} 
\]
is the value of $x$ for which the$\,$ integral has the value$\ \,\alpha $,
as illustrated by the following:

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

\begin{quotation}
$\limfunc{WeibullInv}\left( .73;.5,.3\right) =.3\left( \ln \frac{1}{1-.73}%
\right) ^{\frac{1}{.5}}=\,.51431\,$

$\func{WeibullDist}\left( .51431;.5,.3\right)
=1-e^{-.51431^{.5}.3^{-.5}}=\,.73$
\end{quotation}

\ 

The following plots show cumulative distribution functions $\func{WeibullDist%
}\left( x;a,b\right) $ and density functions $\limfunc{WeibullDen}(x;a,b)$,
for the parameters $(a,b)=(.5,1),(1,1),$ $(3,.5),(3,1)$ and $0\leq x\leq 3$.

\begin{center}
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\TEXUX{$e^{-u}$};linecolor "green";linestyle 1;linethickness 1;pointstyle
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\TEXUX{$3u^{2}e^{-u^{3}}$};linecolor "red";linestyle 1;linethickness
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45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{x};var1name
\TEXUX{$x$};}}

\strut
\end{center}

\subsection{\ Gamma\label{Gamma distribution}}

The \textsl{Gamma distribution} is 
\index{Gamma distribution@Gamma distribution}defined for $x>0$ by the
integral 
\[
\limfunc{GammaDist}\left( x;a,b\right) =\frac{1}{b^{a}\Gamma \left( a\right) 
}\int_{0}^{x}u^{a-1}e^{-\frac{u}{b}}du 
\]
where $\Gamma \left( t\right) =\int_{0}^{\infty }e^{-u}u^{t-1}du$ is the
Gamma function. The parameters $a$ and $b$ are called the shape parameter
and scale parameter, respectively. The mean of this distribution is $ab$ and
the variance is $ab^{2}$. The probability density function for the gamma
distribution is 
\[
\limfunc{GammaDen}(u;a,b)=\frac{1}{b^{a}\Gamma \left( a\right) }u^{a-1}e^{-%
\frac{u}{b}} 
\]

The following plots show cumulative distribution functions $\limfunc{%
GammaDist}\left( x;a,b\right) $ and probability density functions $\limfunc{%
GammaDen}(x;a,b)$ for $(a,b)=\left( 1,.5\right) ,\left( 1,1\right) ,\left(
2,1\right) $ and $0\leq x\leq 4$.

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1;pointstyle "point";function \TEXUX{$e^{-u}$};linecolor "green";linestyle
1;linethickness 1;pointstyle "point";function \TEXUX{$ue^{-u}$};linecolor
"red";linestyle 1;linethickness 1;pointstyle "point";xmin "0";xmax
"4";xviewmin "-0";xviewmax "4";yviewmin "0";yviewmax
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\TEXUX{$\int_{0}^{u}ue^{-u}\,du$};linecolor "red";linestyle 1;linethickness
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49;axesstyle "normal";xis \TEXUX{u};var1name \TEXUX{$x$};}}
\end{center}

\vspace{1pt}\ 

\subsection{\ Beta\label{Beta distribution}}

The \textsl{Beta distribution} is 
\index{Beta distribution@Beta distribution}defined for $0\leq x\leq 1$ by
the integral 
\[
\limfunc{BetaDist}\left( x;v,w\right) =\frac{1}{B\left( v,w\right) }%
\int_{0}^{x}u^{v-1}\left( 1-u\right) ^{w-1}du 
\]
where $B(v,w)=\int_{0}^{1}u^{v-1}\left( 1-u\right) ^{w-1}du$ is the Beta
function with parameters $v$ and $w$.

The probability density function for the beta distribution is 
\[
\limfunc{BetaDen}\left( u;v,w\right) =\frac{u^{v-1}\left( 1-u\right) ^{w-1}}{%
B\left( v,w\right) } 
\]
The parameters $v$ and $w$ are positive real numbers called \textsl{shape
parameters,} and $0\leq u\leq 1$. The mean of the beta distribution is $%
\dfrac{v}{v+w}$.

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

\begin{quotation}
$\limfunc{BetaDist}\left( .5;2,3\right) =\,.6875$

$\limfunc{BetaDen}\left( .5;2,3\right) =\,1.5$
\end{quotation}

\ 

The following plots show cumulative distribution functions $\limfunc{BetaDist%
}\left( x;b,c\right) $ and probability density functions $\limfunc{BetaDen}%
(x;b,c)$ for $(b,c)=\left( 2,3\right) ,\;\left( 5,1\right) ,\;\left(
3,8\right) ,$ and $0\leq x\leq 1$.

\begin{center}
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\special{language "Scientific Word";type "MAPLEPLOT";width 2.1577in;height
1.7573in;depth 0in;display "PICT";function \TEXUX{$\allowbreak 12u\left(
u-1\right) ^{2}$};linecolor "blue";linestyle 1;linethickness 1;pointstyle
"point";function \TEXUX{$5u^{4}$};linecolor "green";linestyle
1;linethickness 1;pointstyle "point";function \TEXUX{$-360u^{2}\left(
u-1\right) ^{7}$};linecolor "red";linestyle 1;linethickness 1;pointstyle
"point";xmin "0";xmax "1";xviewmin "0";xviewmax "1";yviewmin "0";yviewmax
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3x^{4}-8x^{3}+6x^{2}$};linecolor "blue";linestyle 1;linethickness
1;pointstyle "point";function \TEXUX{$x^{5}$};linecolor "green";linestyle
1;linethickness 1;pointstyle "point";function
\TEXUX{$-36x^{10}+120x^{3}+280x^{9}-945x^{8}+1800x^{7}-\allowbreak
2100x^{6}+1512x^{5}-630x^{4}$};linecolor "red";linestyle 1;linethickness
1;pointstyle "point";xmin "0";xmax "1";xviewmin "-0.02";xviewmax
"1.02";yviewmin "-0.02";yviewmax "1.02";rangeset"X";phi 45;theta 45;plottype
4;numpoints 49;axesstyle "normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}

\strut
\end{center}

\subsection{\ Cauchy\label{Cauchy distribution}}

The \textsl{Cauchy cumulative distribution function} is 
\index{Cauchy distribution@Cauchy distribution}defined for all real numbers $%
\alpha $, and for positive $\beta $, by the integral 
\[
\func{CauchyDist}(x;\alpha ,\beta )=\frac{1}{\pi \beta }\int_{-\infty
}^{x}\left( 1+\left( \frac{u-\alpha }{\beta }\right) ^{2}\right) ^{-1}du 
\]
of the \textsl{Cauchy probability density function} 
\[
\limfunc{CauchyDen}(u;\alpha ,\beta )=\frac{1}{\pi \beta \left( 1+\left( 
\frac{u-\alpha }{\beta }\right) ^{2}\right) } 
\]
The median of this distribution is $\alpha $. The Cauchy probability density
function is symmetric about $\alpha $ and has a unique maximum at $\alpha $.

The following plots show probability density functions $\func{CauchyDen}%
(x;\alpha ,\beta )$ and cumulative distribution functions $\func{CauchyDist}%
(x;\alpha ,\beta )$, for the parameters $\left( \alpha ,\beta \right)
=(-3,1) $, $(0,1.5)$, and $(3,1)$, and $-5\leq x\leq 5.$

\begin{center}
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\left( 1+\left( u+3\right) ^{2}\right) }$};linecolor "blue";linestyle
1;linethickness 1;pointstyle "point";function \TEXUX{$\frac{1}{\pi \left(
1+\left( u-3\right) ^{2}\right) }$};linecolor "green";linestyle
1;linethickness 1;pointstyle "point";function \TEXUX{$\frac{.\,66667}{\pi
\left( 1+.\,44444u^{2}\right) }$};linecolor "red";linestyle 1;linethickness
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"5.204";yviewmin "-0.001371";yviewmax "0.3247";rangeset"X";phi 45;theta
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distribution functions}}{}{}{\special{language "Scientific Word";type
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\TEXUX{$\frac{1}{\pi }\int_{-\infty }^{u}\frac{1}{1+\left( u+3\right)
^{2}}\,du$};linecolor "blue";linestyle 1;linethickness 1;pointstyle
"point";function \TEXUX{$\frac{1}{\pi }\int_{-\infty }^{u}\frac{1}{1+\left(
u-3\right) ^{2}}\,du$};linecolor "green";linestyle 1;linethickness
1;pointstyle "point";function \TEXUX{$\frac{.\,66667}{\pi }\int_{-\infty
}^{u}\frac{1}{1+.\,44444u^{2}}\,du$};linecolor "red";linestyle
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"-5.2";xviewmax "5.204";yviewmin "0.02117";yviewmax "0.9792";rangeset"X";phi
45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis
\TEXUX{u};var1name \TEXUX{$x$};}}

\strut
\end{center}

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

The \textsl{cumulative distribution function} $%
\func{UniformDist}\left( x;a,b\right) $ for $a<b$ is the function 
\[
\func{UniformDist}\left( x;a,b\right) =\left\{ 
\begin{array}{ccc}
0 & if & x\leq a \\ 
\frac{x-a}{b-a} & if & a\leq x\leq b \\ 
1 & if & b\leq x
\end{array}
\right. 
\]
The \textsl{probability density function} of the uniform distribution on an
interval $[a,b],$ where $a<b,$ is the function 
\[
\limfunc{UniformDen}(x;a,b)=\left\{ 
\begin{array}{ccc}
0 & if & x\leq a \\ 
\frac{1}{b-a} & if & a\leq x\leq b \\ 
0 & if & b\leq x
\end{array}
\right. 
\]
The uniform random variable is the continuous version of ``choosing a number
at random.'' The probability that a uniform random variable on $\left[
a,b\right] $ will have a value in either of two subintervals of $\left[
a,b\right] $ of equal length is the same.

The following plots show cumulative distribution functions $\func{UniformDist%
}\left( x;a,b\right) $ and probability density functions $\limfunc{UniformDen%
}(x;a,b)$ for $(a,b)=\left( 0,1\right) $, $\left( 1.5,5\right) $, $\left(
3,15\right) $ and $-5\leq x\leq 20$.

\begin{center}
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0,0,0,1,1,1,1,0\right) $};linecolor "blue";linestyle 1;linethickness
1;pointstyle "point";function \TEXUX{$\left(
1.5,0,1.5,.28571,5,.28571,5,0\right) $};linecolor "green";linestyle
1;linethickness 1;pointstyle "point";function \TEXUX{$\left(
3,0,3,\frac{1}{12},15,\frac{1}{12},15,0\right) $};linecolor "red";linestyle
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"-0.3";xviewmax "15.31";yviewmin "-0.02";yviewmax "1.02";rangeset"X";phi
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\TEXUX{$\left( 1.5,0,5,1,20,1\right) $};linecolor "green";linestyle
1;linethickness 1;pointstyle "point";function \TEXUX{$\left(
3,0,15,1,20,1\right) $};linecolor "red";linestyle 1;linethickness
1;pointstyle "point";xmin "0";xmax "20";xviewmin "-0.4";xviewmax
"20.41";yviewmin "-0.02";yviewmax "1.02";rangeset"X";phi 45;theta
45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{x};var1name
\TEXUX{$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 Discrete Distributions}{}{}{DM11-6.tex#Discrete
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}
