Distributions.tex

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\chapter{Additional probability distributions}
\label{ch:distributions}

\section{Beta distribution}
\label{sec:betadist} Table \ref{tb:betadist} shows the properties
of the beta distribution.
\begin{table}[h]
  \centering
  \caption{Properties of the beta distribution}\label{tb:betadist}
\vspace{1 ex}
\begin{tabular}{|l|c|} \hline
  \vbox to 3ex{}Range of random variable & $\left[0,1\right]$\\ *[1ex] \hline
  \vbox to 4ex{}Probability density function & $\displaystyle P\left(x\right)={1\over B\left(\alpha_1,\alpha_2\right)}
  x^{\alpha_1 - 1}\left(1-x\right)^{\alpha_2 - 1}$ \\*[2ex]  \hline
  \vbox to 3ex{}Parameters & $0<\alpha_1<+\infty$ \\
  & $0<\alpha_2<+\infty$\\*[1ex]  \hline
  \vbox to 4ex{}Distribution function & $\displaystyle F\left(x\right)=B\left(x;\alpha_1,\alpha_2\right)$ \\
  &(\cf section \ref{sec:incbeta}) \\*[1ex]  \hline
  \vbox to 3ex{}Average & $\displaystyle{\alpha_1 \over \alpha_1+\alpha_2}$ \\*[2ex] \hline
  \vbox to 4ex{}Variance & $\displaystyle{\alpha_1 \alpha_2
   \over \left(\alpha_1+\alpha_2\right)^2\left(\alpha_1+\alpha_2+1\right)}$ \\*[3ex] \hline
  \vbox to 4ex{}Skewness & $\displaystyle 2{\alpha_1 - \alpha_2
   \over \left(\alpha_1+\alpha_2+2\right)}$ \\*[3ex] \hline
  \vbox to 4ex{}Kurtosis & \parbox{8cm}{$$\begin{array}{l}\displaystyle{\root 3 \of {\alpha_1+\alpha_2+2
  \over \alpha_1 \alpha_2}} \left\{ {\left(\alpha_1+\alpha_2+1\right)
  \over \alpha_1 \alpha_2
  \left(\alpha_1+\alpha_2+2\right)
  \left(\alpha_1+\alpha_2+3\right)}\right.\\*[4ex]
  \qquad \times \left.\left[2\left(\alpha_1+\alpha_2\right)^2+\alpha_1 \alpha_2
  \left(\alpha_1+\alpha_2-6\right)\right]-1\right\}\end{array}$$} \\*[1ex] \hline
\end{tabular}
\end{table}
If the parameters $\alpha_1$ and $\alpha_2$ are equal to 1 the
beta distribution is identical to the uniform distribution over
the interval $\left[0,1\right]$.

The beta distribution is an ad-hoc distribution, which can take
many shapes. Figure \ref{fig:betaDistr} shows a few characteristic
shapes for several values of the parameters. The random variable
is limited to the interval $\left[0,1\right]$. However, any
distribution of a random variable varying over a finite interval
can be mapped to a beta distribution with a suitable variable
transformation.
\begin{figure}
\centering\includegraphics[width=12cm]{Figures/BetaDistribution}
\caption{Many shapes of the beta
distribution}\label{fig:betaDistr}
\end{figure}


\subsection{Beta distribution --- Smalltalk  implementation}
Listing \ref{ls:betadist} shows the implementation of the beta
distribution in Smalltalk.

\begin{listing} Smalltalk implementation of the beta distribution \label{ls:betadist}
\input{Smalltalk/Statistics/DhbBetaDistribution}
\end{listing}


\section{Cauchy distribution}
\label{sec:cauchydist} Table \ref{tb:cauchydist} shows the
properties of the Cauchy distribution. Physicists use the Cauchy
distribution under the name Breit-Wigner or resonnance curve. All
moments of order greater than 0 are not defined as the
corresponding integrals diverge.

\begin{table}[h]
  \centering
  \caption{Properties of the Cauchy distribution}\label{tb:cauchydist}
\vspace{1 ex}
\begin{tabular}{|l|c|} \hline
  \vbox to 3ex{}Range of random variable & $\left]-\infty,+\infty\right[$\\ *[1ex] \hline
  \vbox to 4ex{}Probability density function & $\displaystyle P\left(x\right)=
  {\beta \over \pi \left[\left(x-\mu\right)^2+\beta^2\right]}$ \\*[2ex]  \hline
  \vbox to 3ex{}Parameters & $-\infty<\mu<+\infty$ \\
  & $0<\beta<+\infty$\\*[1ex]  \hline
  \vbox to 4ex{}Distribution function & $\displaystyle F\left(x\right)=
  {1\over 2}+{1\over\pi}\arctan\left({x-\mu\over\beta}\right)$ \\*[1ex]  \hline
  \vbox to 2ex{}Average & (undefined) \\*[1ex] \hline
  \vbox to 2ex{}Variance & (undefined) \\*[1ex] \hline
  \vbox to 2ex{}Skewness & (undefined) \\*[1ex] \hline
  \vbox to 2ex{}Kurtosis & (undefined) \\*[1ex] \hline
\end{tabular}
\end{table}
Figure \ref{fig:cauchyDistr} shows the shapes taken by the Cauchy
distribution for a few values of the parameters. These parameter
are identical to the parameters of the normal distributions shown
in figure \ref{fig:normDistr} so that the reader can compare them.
\begin{figure}
\centering\includegraphics[width=12cm]{Figures/CauchyDistribution}
\caption{Cauchy distribution for a few
parameters}\label{fig:cauchyDistr}
\end{figure}

\subsection{Cauchy distribution --- Smalltalk  implementation}
Listing \ref{ls:cauchydist} shows the implementation of the Cauchy
distribution in Smalltalk.

This implementation returns $\mu$ for the average although the
average is not defined mathematically. Other moment related
quantities are returning {\tt nil}.

\begin{listing} Smalltalk implementation of the Cauchy distribution \label{ls:cauchydist}
\input{Smalltalk/Statistics/DhbCauchyDistribution}
\end{listing}


\section{Exponential distribution}
Table \ref{tb:exponentialdist} shows the properties of the
exponential distribution.
\begin{table}[h]
  \centering
  \caption{Properties of the exponential distribution}\label{tb:exponentialdist}
\vspace{1 ex}
\begin{tabular}{|l|c|} \hline
  \vbox to 3ex{}Range of random variable & $\left[0,+\infty\right[$\\ *[1ex] \hline
  \vbox to 4ex{}Probability density function & $\displaystyle P\left(x\right)={1\over\beta}
  e^{-{x\over\beta}}$ \\*[2ex]  \hline
  \vbox to 3ex{}Parameters & $0<\beta<+\infty$ \\*[1ex]  \hline
  \vbox to 4ex{}Distribution function & $\displaystyle F\left(x\right)=1-e^{-{x\over\beta}}$ \\*[1ex]  \hline
  \vbox to 3ex{}Average & $\beta$ \\*[1ex] \hline
  \vbox to 3ex{}Variance & $\beta^2$ \\*[1ex] \hline
  \vbox to 3ex{}Skewness & $2$ \\*[1ex] \hline
  \vbox to 3ex{}Kurtosis & $6$ \\*[1ex] \hline
\end{tabular}
\end{table}

The exponential distribution describes the distribution of the
time of occurrence between independent random events with a
constant probability of occurrence. It is used in queuing theory
and in nuclear physics. Figure \ref{fig:expDistr} shows the shapes
taken by the exponential distribution for a few values of the
parameters.
\begin{figure}
\centering\includegraphics[width=12cm]{Figures/ExponentialDistribution}
\caption{Exponential distribution for a few
parameters}\label{fig:expDistr}
\end{figure}

\subsection{Exponential distribution --- Smalltalk  implementation}
Listing \ref{ls:exponentialdist} shows the implementation of the
exponential distribution in Smalltalk.

\begin{listing} Smalltalk implementation of the exponential distribution \label{ls:exponentialdist}
\input{Smalltalk/Statistics/DhbExponentialDistribution}
\end{listing}


\section{Fisher-Tippett distribution}
\label{sec:fishertippettdist} Table \ref{tb:fishertippettdist}
shows the properties of the fishertippett distribution.
\begin{table}[h]
  \centering
  \caption{Properties of the Fisher-Tippett distribution}\label{tb:fishertippettdist}
\vspace{1 ex}
\begin{tabular}{|l|c|} \hline
  \vbox to 3ex{}Range of random variable & $\left]-\infty,+\infty\right[$\\ *[1ex] \hline
  \vbox to 4ex{}Probability density function & $\displaystyle P\left(x\right)=
  {1\over\beta}e^{-{x-\alpha\over\beta}-e^{-{x-\alpha\over\beta}}}$ \\*[2ex]  \hline
  \vbox to 3ex{}Parameters & $-\infty<\alpha<+\infty$ \\
  & $0<\beta<+\infty$\\*[1ex]  \hline
  \vbox to 4ex{}Distribution function & $\displaystyle F\left(x\right)=e^{-e^{-{x-\alpha\over\beta}}}$ \\*[1ex]  \hline
  \vbox to 3ex{}Average & $\alpha+\gamma\beta$ \\*[1ex] \hline
  \vbox to 4ex{}Variance & $\displaystyle{\pi\beta\over\sqrt{6}}$ \\*[2ex] \hline
  \vbox to 3ex{}Skewness & $1.3$ \\*[1ex] \hline
  \vbox to 3ex{}Kurtosis & $2.4$ \\*[1ex] \hline
\end{tabular}
\end{table}
In this table $\gamma=0.5772156649\ldots$ is the Euler constant.

The Fisher-Tippett distribution describes the distribution of
extreme values. Figure \ref{fig:ftippettDistr} shows the shapes
taken by the Fisher-Tippett distribution for a few values of the
parameters. These parameter are identical to the parameters of the
normal distributions shown in figure \ref{fig:normDistr} so that
the reader can compare them.
\begin{figure}
\centering\includegraphics[width=12cm]{Figures/FisherTippettDistribution}
\caption{Fisher-Tippett distribution for a few
parameters}\label{fig:ftippettDistr}
\end{figure}

\subsection{Fisher-Tippett distribution --- Smalltalk  implementation}
Listing \ref{ls:fishertippettdist} shows the implementation of the
Fisher-Tippett distribution in Smalltalk.

\begin{listing} Smalltalk implementation of the Fisher-Tippett distribution \label{ls:fishertippettdist}
\input{Smalltalk/Statistics/DhbFisherTippettDistribution}
\end{listing}


\section{Laplace distribution}
\label{sec:laplacedist} Table \ref{tb:laplacedist} shows the
properties of the Laplace distribution.
\begin{table}[h]
  \centering
  \caption{Properties of the Laplace distribution}\label{tb:laplacedist}
\vspace{1 ex}
\begin{tabular}{|l|c|} \hline
  \vbox to 3ex{}Range of random variable & $\left]-\infty,+\infty\right[$\\ *[1ex] \hline
  \vbox to 4ex{}Probability density function & $\displaystyle P\left(x\right)={1\over 2\beta} e^{-{\left|x-\alpha\right|\over\beta}}$ \\*[2ex]  \hline
  \vbox to 3ex{}Parameters & $-\infty<\alpha<+\infty$ \\
  & $0<\beta<+\infty$\\*[1ex]  \hline
  \vbox to 4ex{}Distribution function &
  \parbox{6cm}{$$F\left(x\right)=\left\{
  \begin{array}{ll}
  {1\over 2}e^{-{\alpha-x\over\beta}}&\mbox{\quad for
  $x<\alpha$}\\*[1ex]
  1-{1\over 2}e^{-{x-\alpha\over\beta}}&\mbox{\quad for $x\ge\alpha$}
  \end{array}\right.$$}\\*[1ex]  \hline
  \vbox to 3ex{}Average & $\alpha+\beta$ \\*[1ex] \hline
  \vbox to 3ex{}Variance & $2\beta^2$ \\*[1ex] \hline
  \vbox to 3ex{}Skewness & $0$ \\*[1ex] \hline
  \vbox to 3ex{}Kurtosis & $3$ \\*[1ex] \hline
\end{tabular}
\end{table}
The Laplace distribution is an ad-hoc distribution made of two
exponential distributions, one on each side of the peak. Figure
\ref{fig:laplaceDistr} shows the shapes taken by the Laplace
distribution for a few values of the parameters. These parameter
are identical to the parameters of the normal distributions shown
in figure \ref{fig:normDistr} so that the reader can compare them.
\begin{figure}
\centering\includegraphics[width=12cm]{Figures/LaplaceDistribution}
\caption{Laplace distribution for a few
parameters}\label{fig:laplaceDistr}
\end{figure}

\subsection{Laplace distribution --- Smalltalk  implementation}
Listing \ref{ls:laplacedist} shows the implementation of the
Laplace distribution in Smalltalk.

\begin{listing} Smalltalk implementation of the Laplace distribution \label{ls:laplacedist}
\input{Smalltalk/Statistics/DhbLaplaceDistribution}
\end{listing}


\section{Log normal  distribution}
Table \ref{tb:lognormaldist} shows the properties of the log
normal distribution.
\begin{table}[h]
  \centering
  \caption{Properties of the log normal distribution}\label{tb:lognormaldist}
\vspace{1 ex}
\begin{tabular}{|l|c|} \hline
  \vbox to 3ex{}Range of random variable & $\left[0,+\infty\right[$\\ *[1ex] \hline
  \vbox to 4ex{}Probability density function & $\displaystyle P\left(x\right)=
  {1\over x\sqrt{2\pi\sigma^2}}e^{-{\left(\ln x-\mu\right)^2\over 2\sigma^2}}$ \\*[2ex]  \hline
  \vbox to 3ex{}Parameters & $-\infty<\mu<+\infty$ \\
  & $0<\sigma<+\infty$\\*[1ex]  \hline
  \vbox to 4ex{}Distribution function & (no closed expression) \\*[1ex]  \hline
  \vbox to 3ex{}Average & $e^{\mu+{\sigma^2\over 2}}$ \\*[1ex] \hline
  \vbox to 4ex{}Variance & $e^{2\mu+\sigma^2}\left(e^{\sigma^2}-1\right)$ \\*[1ex] \hline
  \vbox to 4ex{}Skewness & $\sqrt{e^{\sigma^2}-1}\left(e^{\sigma^2}+2\right)$ \\*[2ex] \hline
  \vbox to 3ex{}Kurtosis & $ e^{4\sigma^2}+2e^{3\sigma^2}+3e^{2\sigma^2}-6$ \\*[1ex] \hline
\end{tabular}
\end{table}
The log normal distribution is used to describe quantities that
are the product of a large number of other quantities. It is an
ad-hoc distribution whose shape is similar to that of gamma
distributions with $\alpha>1$. Figure \ref{fig:logNormDistr} shows
the shapes taken by the log normal distribution for a few values
of the parameters.
\begin{figure}
\centering\includegraphics[width=12cm]{Figures/LogNormalDistribution}
\caption{Log normal distribution for a few parameters}\label{fig:logNormDistr}
\end{figure}

\subsection{Log normal distribution --- Smalltalk  implementation}
Listing \ref{ls:lognormaldist} shows the implementation of the log
normal  distribution in Smalltalk.

\begin{listing} Smalltalk implementation of the log normal distribution \label{ls:lognormaldist}
\input{Smalltalk/Statistics/DhbLogNormalDistribution}
\end{listing}


\section{Triangular distribution}
Table \ref{tb:triangdist} shows the properties of the triangular
distribution.
\begin{table}[h]
  \centering
  \caption{Properties of the triangular distribution}\label{tb:triangdist}
\vspace{1 ex}
\begin{tabular}{|l|c|} \hline
  \vbox to 3ex{}Range of random variable & $\left[a,b\right]$\\ *[1ex] \hline
  \vbox to 4ex{}Probability density function & \parbox{6cm}{$$\displaystyle P\left(x\right)=
  \left\{\begin{array}{ll}
  {2\left(x-a\right)\over\left(b-a\right)\left(c-a\right)}&\mbox{\quad if $a\le x\le
  c$,}\\*[3ex]
  {2\left(b-x\right)\over\left(b-a\right)\left(b-c\right)}&\mbox{\quad if $c\le x\le
  b$.}
  \end{array}
  \right.$$} \\*[2ex]  \hline
  \vbox to 3ex{}Parameters & $-\infty<a\le c\le b<+\infty$ \\
  & $a<b$\\*[1ex]  \hline
  \vbox to 4ex{}Distribution function & \parbox{8cm}{$$\displaystyle F\left(x\right)=
  \left\{\begin{array}{ll}
  {\left(x-a\right)^2\over\left(b-a\right)\left(c-a\right)}&\mbox{\quad if $a\le x\le
  c$,}\\*[3ex]
  1-{\left(b-x\right)^2\over\left(b-a\right)\left(b-c\right)}&\mbox{\quad if $c\le x\le
  b$.}
  \end{array}
  \right.$$} \\*[1ex]  \hline
  \vbox to 4ex{}Average & $\displaystyle{a+b+c\over 3}$ \\*[1ex] \hline
  \vbox to 4ex{}Variance & $\displaystyle{a^2+b^2+c^2-ab-ac-bc\over 18}$ \\*[1ex] \hline
  \vbox to 4ex{}Skewness & $\displaystyle{a^3+b^3+c^3\over 135}+\ldots$ \\*[1ex] \hline
  \vbox to 4ex{}Kurtosis & $\ldots$ \\*[1ex] \hline
\end{tabular}
\end{table}
The triangular distribution is ad-hoc distribution used when a
variable is limited to an interval.

\subsection{Triangular distribution --- Smalltalk  implementation}
Listing \ref{ls:triangdist} shows the implementation of the
triangular distribution in Smalltalk.

\begin{listing} Smalltalk implementation of the triangular distribution \label{ls:triangdist}
\input{Smalltalk/Statistics/DhbTriangularDistribution}
\end{listing}


\section{Uniform distribution}
Table \ref{tb:uniformdist} shows the properties of the uniform
distribution.
\begin{table}[h]
  \centering
  \caption{Properties of the uniform distribution}\label{tb:uniformdist}
\vspace{1 ex}
\begin{tabular}{|l|c|} \hline
  \vbox to 3ex{}Range of random variable & $\left[a,b\right]$\\ *[1ex] \hline
  \vbox to 4ex{}Probability density function & $\displaystyle P\left(x\right)={1\over b-a}$ \\*[2ex]  \hline
  \vbox to 3ex{}Parameters & $-\infty<a<b<+\infty$\\*[1ex]  \hline
  \vbox to 4ex{}Distribution function & $\displaystyle F\left(x\right)={x-a\over b-a}$ \\*[1ex]  \hline
  \vbox to 3ex{}Average & ${a+b\over 2}$ \\*[1ex] \hline
  \vbox to 3ex{}Variance & ${\left(b-a\right)^2\over 12}$ \\*[1ex] \hline
  \vbox to 3ex{}Skewness & $0$ \\*[1ex] \hline
  \vbox to 3ex{}Kurtosis & $-1.2$ \\*[1ex] \hline
\end{tabular}
\end{table}
The uniform distribution is another ad-hoc distribution used when
a variable is limited to an interval.

\subsection{Uniform distribution --- Smalltalk  implementation}
Listing \ref{ls:uniformdist} shows the implementation of the
uniform distribution in Smalltalk.

\begin{listing} Smalltalk implementation of the uniform distribution \label{ls:uniformdist}
\input{Smalltalk/Statistics/DhbUniformDistribution}
\end{listing}

\section{Weibull distribution}
Table \ref{tb:weibulldist} shows the properties of the Weibull
distribution.
\begin{table}[h]
  \centering
  \caption{Properties of the Weibull distribution}\label{tb:weibulldist}
\vspace{1 ex}
\begin{tabular}{|l|c|} \hline
  \vbox to 3ex{}Range of random variable & $\left[0,+\infty\right[$\\ *[1ex] \hline
  \vbox to 4ex{}Probability density function & $\displaystyle P\left(x\right)=
  {\alpha x^{\alpha-1}\over \beta^{\alpha}}e^{-\left({x \over \beta}\right)^{\alpha}}$ \\*[2ex]  \hline
  \vbox to 3ex{}Parameters & $0<\alpha<\infty$ \\
  & $0<\beta<\infty$\\*[1ex]  \hline
  \vbox to 4ex{}Distribution function & $\displaystyle F\left(x\right)=
  1-e^{-\left({x \over \beta}\right)^{\alpha}}$ \\*[1ex]  \hline
  \vbox to 3ex{}Average & ${\beta\over\alpha}\Gamma\left({1\over\alpha}\right)$ \\*[1ex] \hline
  \vbox to 4ex{}Variance & ${\beta^2\over\alpha}\left[
  2\Gamma\left({2\over\alpha}\right)-{1\over\alpha}\Gamma\left({1\over\alpha}\right)^2\right]$ \\*[1ex] \hline
  \vbox to 3ex{}Skewness & $ $ \\*[1ex] \hline
  \vbox to 3ex{}Kurtosis & $ $ \\*[1ex] \hline
\end{tabular}
\end{table}
The Weibull distribution is used to model the behavior of
reliability. It is defined by its acceptance function. Its shape
is similar to that of the gamma distribution and, thus, can be
applied to the same types of problems.Figure
\ref{fig:weibullDistr} shows the shapes taken by the Weibull
distribution for a few values of the parameters.
\begin{figure}
\centering\includegraphics[width=12cm]{Figures/WeibullDistribution}
\caption{Weibull distribution for a few
parameters}\label{fig:weibullDistr}
\end{figure}

Because the Weibull distribution is defined by its distribution
function, the estimation of the initial values of the parameters
from a histogram is made by computing the distribution function at
2 positions. These positions are determined using the histogram
limits and the average so that the estimation of the distribution
function using the histogram has enough significance.

\subsection{Weibull distribution --- Smalltalk  implementation}
\label{sec:weibull} Listing \ref{ls:weibulldist} shows the
implementation of the Weibull distribution in Smalltalk.

\begin{listing} Smalltalk implementation of the Weibull distribution \label{ls:weibulldist}
\input{Smalltalk/Statistics/DhbWeibullDistribution}
\end{listing}



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