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%TCIDATA{<META NAME="Title" CONTENT="Distributions and Densities">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 16:50:31 1997}
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\section{\textsl{
\index{Distribution@Distribution}}\label{Probability density function}\label%
{Distribution}Distributions \label{Cumulative distribution function}and
Densities}

A \textsl{cumulative distribution function } 
\index{Cumulative distribution function@Cumulative distribution function}$%
F\left( x\right) $ of a 
\index{Statistics@Statistics!cumulative distributions@cumulative distributions}%
\label{Statistics, cumulative distributions}random variable $X$ is the
function $F\left( x\right) =P\left( X\leq x\right) $, the probability that $%
X\leq x$. If $F(x)$ has a derivative $f(x)$, then $f(x)$ is nonnegative and
is called the 
\index{Probability density function@Probability density function}\textsl{%
probability density function} of $x$. The \textsl{inverse distribution
function} $G(\alpha )$ satisfies $G\left( F\left( x\right) \right) =x$ and $%
F\left( G\left( \alpha \right) \right) =\alpha $. The \textsl{Scientific
Notebook} names for these functions are obtained by adding Dist, Den, or Inv
to the name of the distribution. For example, $%
\func{NormalDist}$, $\func{NormalDen}$, and $\func{NormalInv}$ are the three
functions for the normal distribution. Such function names will
automatically turn gray when typed in mathematics mode.

\subsection{\ Cumulative Distribution Functions}

A \textsl{cumulative distribution function} is a nondecreasing function
defined on the interval $\left( -\infty ,\infty \right) $, with values in
the interval $\left[ 0,1\right] $. The definition of a distribution function
generally describes only the values where the function is positive, the
implicit assumption being that the distribution function is zero up to that
point. For discrete cumulative distribution functions, the definition also
gives only the values where the function changes, the implicit assumption
being that the cumulative distribution function is a step function.
Commonly, definitions of these functions are stated only for integers. The 
\index{Statistics@Statistics!probability density function@probability density function}%
\label{Statistics, probability density function} definition of a density
function also generally describes only the values where the function is
positive, the implicit assumption being that the function is zero elsewhere.

These distribution and density functions satisfy the relationships 
\begin{eqnarray*}
f(x) &=&%
\frac{d}{dx}F(x)\vspace{6pt} \\
F(x) &=&\int_{-\infty }^{x}f(u)du\vspace{6pt}
\end{eqnarray*}
Also note that the cumulative distribution function satisfies $%
\lim_{x\rightarrow \infty }F(x)=1$ and $\lim_{x\rightarrow -\infty }F(x)=0$.
In \textsl{Scientific Notebook}, cumulative distribution functions are named 
$\limfunc{FunctionDist},$ and the density functions are named $\limfunc{%
FunctionDen}$. For example, the probability density functions for the normal
distributions are called $\limfunc{NormalDen}$.

\textsl{Scientific Notebook }includes several families of distributions:
Normal, Cauchy, Student's t, Chi-square, F, Exponential, Weibull, Gamma,
Beta, Uniform, Binomial, Poisson, and Hypergeometric.

\subsection{\ \label{Inverse distribution functions}Inverse Distribution
Functions}

For a distribution function $F$ mapping $\left( -\infty ,\infty \right) $
into $\left[ 0,1\right] $, the \textsl{inverse distribution function} $G$
performs the corresponding inverse mapping from (a subset of) $\left[
0,1\right] $ into $\left( -\infty ,\infty \right) $; that is, $G\left(
F\left( x\right) \right) =x$ and $F\left( G\left( \alpha \right) \right)
=\alpha $. Equivalently, 
\[
\text{Prob}\left[ X\leq G\left( \alpha \right) \right] =F(x)=\alpha 
\]
Note that the value that is \emph{exceeded} with probability $\alpha $ is
given by the function $G(1-\alpha )$. This function is also of interest. 
\[
\begin{array}{c}
\text{Prob}\left[ X\leq G\left( 1-\alpha \right) \right] =F(x) \\ 
\quad \quad =1-\alpha =1-\text{Prob}\left[ X\leq G\left( \alpha \right)
\right]
\end{array}
\]
In \textsl{Scientific Notebook}, when cumulative distribution functions are
named $\limfunc{FunctionDist}$, then the inverse cumulative distribution
functions are named $\limfunc{FunctionInv}$. For example, $\limfunc{NormalInv%
}$ is the name of the inverse cumulative distribution function for the
normal distribution.

\subsection{\ Distribution Tables\label{Distribution tables}}

Depending on the particular family of distributions, the distribution tables
in statistics books list function values for selected parameters of one of
the functions described earlier---either the cumulative distribution, the
inverse cumulative distribution, or the probability density function. With
access to these functions in \textsl{Scientific Notebook}, not only can you
compute the tabular entries easily and accurately, but you can also find the
corresponding values directly for any variables and parameters to any degree
of accuracy you wish. With \textsl{Scientific Notebook,} you no longer need
to rely on books of tables.

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

\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}

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\hyperref{\FRAME{itbpF}{0.5275in}{0.3399in}{0.0519in}{}{}{index.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid-file
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"1";cropbottom "0";filename 'index.wmf';file-properties "XNPEU";}}}{}{}{%
DMIndex.tex#Math Index}\quad {\small Index entries: \hyperref{ Data}{}{}{%
DMIndex.tex#Data}, \hyperref{Distribution}{}{}{DMIndex.tex#Distribution}, 
\hyperref{Matrices}{}{}{DMIndex.tex#Matrices}, \hyperref{Reshape}{}{}{%
DMIndex.tex#Reshape}, \hyperref{Statistics}{}{}{DMIndex.tex#Statistics}}

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