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<html>
<head>
<title>
PROB - Probability Density Functions
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
PROB <br> Probability Density Functions
</h1>
<hr>
<p>
<b>PROB</b>
is a FORTRAN90 library which
handles various discrete and
continuous probability density functions
("PDF's").
</p>
<p>
For a discrete variable X, PDF(X) is the probability that the value
X will occur; for a continuous variable, PDF(X) is the probability
density of X, that is, the probability of a value between X and X+dX
is PDF(X) * dX.
</p>
<p>
The corresponding cumulative density functions or "CDF"'s are also
handled. For a discrete or continuous variable, CDF(X) is the
probability that the variable takes on a value less than or equal to X.
</p>
<p>
In some cases, the inverse of the CDF can easily be computed.
If
<pre><b>
X = CDF_INV ( P )
</b></pre>
then we are asserting that the value <b>X</b> has a cumulative
probability density function of <b>P</b>, in other words,
the probability that the variable is less than or equal to <b>X</b>
is <b>P</b>. If the CDF cannot be analytically inverted, there
are simple ways to try to estimate the inverse. Depending on
the PDF, these methods may be rapid and accurate, or not.
</p>
<p>
For most distributions, the <i>mean</i> or "average value" or
"expected value" is also available. For a discrete variable, MEAN
is simply the sum of the products X * PDF(X); for a continuous
variable, MEAN is the integral of X * PDF(X) over the range.
For the distributions covered here, the means are known beforehand,
and no summation or integration is required.
</p>
<p>
For most distributions, the <i>variance</i> is available. For a
discrete variable, the variance is the sum of the products
( X - MEAN )**2 * PDF(X); for a continuous variable, the
variance is the integral of ( X - MEAN )**2 * PDF(X) over the range.
The square root of the variance is known as the <i>standard
deviation</i>. For the distributions covered here, the variances are
often known beforehand, and no summation or integration is required.
</p>
<p>
For many of the distributions, it is possible to repeatedly
request "samples", that is, a pseudorandom sequence of realizations
of the PDF. These samples are always associated with an integer
seed, which controls the calculation. Using the same seed as input
will guarantee the same sample value on output. Ultimately, a
random number generator must be invoked internally. In most cases,
the current code will call a routine called <b>R8_UNIFORM_01</b>,
which uses a very basic, and old, random number generator.
You may prefer a different random number generator
for this purpose.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>PROB</b> is available in
<a href = "../../cpp_src/prob/prob.html">a C++ version</a> and
<a href = "../../f_src/prob/prob.html">a FORTRAN90 version</a> and
<a href = "../../m_src/prob/prob.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/asa005/asa005.html">
ASA005</a>,
a FORTRAN90 library which
evaluates the CDF of the noncentral T distribution.
</p>
<p>
<a href = "../../f_src/asa066/asa066.html">
ASA066</a>,
a FORTRAN90 library which
evaluates the CDF of the normal distribution.
</p>
<p>
<a href = "../../f_src/asa076/asa076.html">
ASA076</a>,
a FORTRAN90 library which
evaluates the Owen's T function, needed to compute the CDF of the
noncentral T distribution.
</p>
<p>
<a href = "../../f_src/asa091/asa091.html">
ASA091</a>,
a FORTRAN90 library
which evaluates the percentage points of the Chi-Squared distribution.
</p>
<p>
<a href = "../../f_src/asa111/asa111.html">
ASA111</a>,
a FORTRAN90 library which
evaluates the percentage points of the normal distribution.
</p>
<p>
<a href = "../../f_src/asa152/asa152.html">
ASA152</a>,
a FORTRAN90 library which
evaluates point and cumulative probabilities associated with the
hypergeometric distribution;
this is Applied Statistics Algorithm 152;
</p>
<p>
<a href = "../../f_src/asa226/asa226.html">
ASA226</a>,
a FORTRAN90 library which
evaluates the CDF of the noncentral Beta distribution.
</p>
<p>
<a href = "../../f_src/asa241/asa241.html">
ASA241</a>,
a FORTRAN90 library which
evaluates the percentage points of the normal distribution.
</p>
<p>
<a href = "../../f_src/asa243/asa243.html">
ASA243</a>,
a FORTRAN90 library which
evaluates the CDF of the noncentral T distribution.
</p>
<p>
<a href = "../../f_src/asa266/asa266.html">
ASA266</a>,
a FORTRAN90 library which
evaluates various properties of the Dirichlet probability density function;
this is Applied Statistics Algorithm 266;
</p>
<p>
<a href = "../../f_src/asa310/asa310.html">
ASA310</a>,
a FORTRAN90 library which
computes the CDF of the noncentral Beta distribution.
</p>
<p>
<a href = "../../f_src/beta_nc/beta_nc.html">
BETA_NC</a>,
a FORTRAN90 library which
evaluates the CDF of the noncentral Beta distribution.
</p>
<p>
<a href = "../../f_src/dcdflib/dcdflib.html">
DCDFLIB</a>,
a FORTRAN90 library which
evaluates and inverts a number of statistical distributions.
</p>
<p>
<a href = "../../f_src/discrete_pdf_sample/discrete_pdf_sample.html">
DISCRETE_PDF_SAMPLE</a>,
a FORTRAN90 program which
demonstrates how to construct a Probability Density Function (PDF)
from a table of sample data, and then to use that PDF to create new samples.
</p>
<p>
<a href = "../../cpp_src/gsl/gsl.html">
GSL</a>,
a C++ library which
includes many routines for evaluating
probability distributions.
</p>
<p>
<a href = "../../f_src/normal/normal.html">
NORMAL</a>,
a FORTRAN90 library which
samples the normal distribution.
</p>
<p>
<a href = "../../f_src/random_data/random_data.html">
RANDOM_DATA</a>,
a FORTRAN90 library which
generates sample points for
various probability distributions, spatial dimensions, and geometries;
</p>
<p>
<a href = "../../f_src/test_values/test_values.html">
TEST_VALUES</a>,
a FORTRAN90 library which
contains sample values for a number of distributions.
</p>
<p>
<a href = "../../f77_src/toms441/toms441.html">
TOMS441</a>,
a FORTRAN77 library which
samples the dipole distribution;<br>
this is ACM TOMS algorithm 441.
</p>
<p>
<a href = "../../f_src/uniform/uniform.html">
UNIFORM</a>,
a FORTRAN90 library which
samples the uniform distribution.
</p>
<p>
<a href = "../../f_src/ziggurat/ziggurat.html">
ZIGGURAT</a>,
a FORTRAN90 program which
generates points from a uniform, normal or exponential distribution, using
the ziggurat method.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Roger Abernathy, Robert Smith,<br>
Algorithm 724:
Program to Calculate F Percentiles,<br>
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</li>
<li>
Milton Abramowitz, Irene Stegun,<br>
Handbook of Mathematical Functions,<br>
National Bureau of Standards, 1964,<br>
ISBN: 0-486-61272-4,<br>
LC: QA47.A34.
</li>
<li>
AG Adams,<br>
Algorithm 39:
Areas Under the Normal Curve,<br>
Computer Journal, <br>
Volume 12, 1969, pages 197-198.
</li>
<li>
Joachim Ahrens, Ulrich Dieter,<br>
Generating Gamma Variates by a Modified Rejection Technique,<br>
Communications of the ACM, <br>
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</li>
<li>
Joachim Ahrens, Ulrich Dieter,<br>
Computer Methods for Sampling from Gamma, Beta, Poisson and
Binomial Distributions.<br>
Computing,<br>
Volume 12, 1974, pages 223-246.
</li>
<li>
Joachim Ahrens, Klaus-Dieter Kohrt, Ulrich Dieter,<br>
Algorithm 599:
Sampling from Gamma and Poisson Distributions,<br>
ACM Transactions on Mathematical Software,<br>
Volume 9, Number 2, June 1983, pages 255-257.
</li>
<li>
Jerry Banks, editor,<br>
Handbook of Simulation,<br>
Wiley, 1998,<br>
ISBN: 0471134031,<br>
LC: T57.62.H37.
</li>
<li>
JD Beasley, SG Springer,<br>
Algorithm AS 111:
The Percentage Points of the Normal Distribution,<br>
Applied Statistics,<br>
Volume 26, 1977, pages 118-121.
</li>
<li>
Frank Benford,<br>
The Law of Anomalous Numbers,<br>
Proceedings of the American Philosophical Society,<br>
Volume 78, 1938, pages 551-572.
</li>
<li>
Jose Bernardo,<br>
Algorithm AS 103:
Psi ( Digamma ) Function,<br>
Applied Statistics,<br>
Volume 25, Number 3, 1976, pages 315-317.
</li>
<li>
Donald Best, Nicholas Fisher,<br>
Efficient Simulation of the von Mises Distribution,<br>
Applied Statistics,<br>
Volume 28, Number 2, pages 152-157.
</li>
<li>
Donald Best, Roberts,<br>
Algorithm AS 91:
The Percentage Points of the Chi-Squared Distribution,<br>
Applied Statistics,<br>
Volume 24, Number 3, 1975, pages 385-390.
</li>
<li>
Paul Bratley, Bennett Fox, Linus Schrage,<br>
A Guide to Simulation,<br>
Second Edition,<br>
Springer, 1987,<br>
ISBN: 0387964673,<br>
LC: QA76.9.C65.B73.
</li>
<li>
William Cody,<br>
An Overview of Software Development for Special Functions,
in Numerical Analysis Dundee, 1975, <br>
edited by GA Watson,<br>
Lecture Notes in Mathematics, 506, <br>
Springer, 1976,<br>
LC: QA3.L28.v506.
</li>
<li>
William Cody,<br>
Rational Chebyshev Approximations for the Error Function,<br>
Mathematics of Computation,<br>
Volume 23, Number 107, July 1969, pages 631-638.
</li>
<li>
William Cody, Kenneth Hillstrom,<br>
Chebyshev Approximations for the Natural Logarithm of the
Gamma Function,
Mathematics of Computation,<br>
Volume 21, Number 98, April 1967, pages 198-203.
</li>
<li>
BE Cooper,<br>
Algorithm AS 5:
The Integral of the Non-Central T-Distribution,<br>
Applied Statistics,<br>
Volume 17, 1968, page 193.
</li>
<li>
Luc Devroye,<br>
Non-Uniform Random Variate Generation,<br>
Springer, 1986,<br>
ISBN: 0387963057,<br>
LC: QA274.D48
</li>
<li>
Merran Evans, Nicholas Hastings, Brian Peacock,<br>
Statistical Distributions,<br>
Wiley, 2000,<br>
ISBN: 0471371246,<br>
LC: QA273.6E92.
</li>
<li>
Nicholas Fisher,<br>
Statistical Analysis of Circular Data,<br>
Cambridge, 1993,<br>
ISBN: 0521568900,<br>
LC: QA276.F488
</li>
<li>
Nicholas Fisher, Toby Lewis, Brian Embleton,<br>
Statistical Analysis of Spherical Data,<br>
Cambridge, 2003,<br>
ISBN13: 978-0521456999,<br>
LC: QA276.F489
</li>
<li>
Darren Glass, Philip Lowry,<br>
Quasigeometric Distributions and Extra Inning Baseball Games,<br>
Mathematics Magazine,<br>
Volume 81, Number 2, April 2008, pages 127-137.
</li>
<li>
John Hart, Ward Cheney, Charles Lawson, Hans Maehly,
Charles Mesztenyi, John Rice, Henry Thatcher,
Christoph Witzgall,<br>
Computer Approximations,<br>
Wiley, 1968,<br>
LC: QA297.C64.
</li>
<li>
Geoffrey Hill,<br>
Algorithm 518:
Incomplete Bessel Function I0: The Von Mises Distribution,<br>
ACM Transactions on Mathematical Software,<br>
Volume 3, Number 3, September 1977, pages 279-284.
</li>
<li>
Ted Hill,<br>
The First Digit Phenomenon,<br>
American Scientist,<br>
Volume 86, July/August 1998, pages 358-363.
</li>
<li>
Mark Johnson,<br>
Multivariate Statistical Simulation:
A Guide to Selecting and Generating Continuous Multivariate
Distributions,<br>
Wiley, 1987,<br>
ISBN: 0471822906,<br>
LC: QA278.J62
</li>
<li>
Norman Johnson, Samuel Kotz, Narayanaswamy Balakrishnan,<br>
Continuous Univariate Distributions,<br>
Second edition,<br>
Wiley, 1994,<br>
ISBN: 0471584940,<br>
LC: QA273.6.J6
</li>
<li>
Norman Johnson, Samuel Kotz, Adrienne Kemp,<br>
Univariate Discrete Distributions,<br>
Third edition,<br>
Wiley, 2005,<br>
ISBN: 0471272469,<br>
LC: QA273.6.J64
</li>
<li>
William Kennedy, James Gentle,<br>
Statistical Computing,<br>
Marcel Dekker, 1980,<br>
ISBN: 0824768981,<br>
LC: QA276.4 K46.
</li>
<li>
Robert Knop,<br>
Algorithm 441:
Random Deviates from the Dipole Distribution,<br>
ACM Transactions on Mathematical Software,<br>
Volume 16, Number 1, January 1973, page 51.
</li>
<li>
Kalimutha Krishnamoorthy,<br>
Handbook of Statistical Distributions with Applications,<br>
Chapman and Hall, 2006,<br>
ISBN: 1-58488-635-8,<br>
LC: QA273.6.K75.
</li>
<li>
Henry Kucera, Winthrop Francis,<br>
Computational Analysis of Present-Day American English,<br>
Brown University Press, 1967,<br>
ISBN13: 978-0870571053,<br>
LC: PE2839.K8.
</li>
<li>
Kenneth Lange,<br>
Mathematical and Statistical Methods for Genetic Analysis,<br>
Springer, 1997,<br>
ISBN: 0387953892,<br>
LC: QH438.4.M33.L36.
</li>
<li>
Alfred Lotka,<br>
The frequency distribution of scientific productivity,<br>
Journal of the Washington Academy of Sciences,<br>
Volume 16, Number 12, 1926, pages 317-324.
</li>
<li>
KL Majumder, GP Bhattacharjee,<br>
Algorithm AS 63:
The incomplete Beta Integral,<br>
Applied Statistics,<br>
Volume 22, number 3, 1973, pages 409-411.
</li>
<li>
Kanti Mardia, Peter Jupp,<br>
Directional Statistics,<br>
Wiley, 2000,<br>
ISBN: 0471953334,<br>
LC: QA276.M335
</li>
<li>
<a href = "http://www.geocities.com/~mikemclaughlin/math_stat/Dists/Compendium.html">
Michael McLaughlin</a><br>
A Compendium of Common Probability Distributions
</li>
<li>
Paul Nahin,<br>
Digital Dice: Computational Solutions to Practical Probability Problems,<br>
Princeton University Press, 2008,<br>
ISBN13: 978-0-691-12698-2,<br>
LC: QA273.25.N34.
</li>
<li>
Keith Ord,<br>
Families of Frequency Distributions,<br>
Lubrecht & Cramer, 1972,<br>
ISBN: 0852641370,<br>
LC: QA273.6.O7.
</li>
<li>
Donald Owen,<br>
Tables for Computing Bivariate Normal Probabilities,<br>
The Annals of Mathematical Statistics,<br>
Volume 27, Number 4, December 1956, pages 1075-1090.
</li>
<li>
Frank Powell,<br>
Statistical Tables for Sociology, Biology and Physical Sciences,<br>
Cambridge University Press, 1982,<br>
ISBN: 0521284732,<br>
LC: QA276.25.S73.
</li>
<li>
Sudarshan Raghunathan,<br>
Making a Supercomputer Do What You Want: High Level Tools for
Parallel Programming,<br>
Computing in Science and Engineering,<br>
Volume 8, Number 5, September/October 2006, pages 70-80.
</li>
<li>
Ralph Raimi,<br>
The Peculiar Distribution of First Digits,<br>
Scientific American,<br>
December 1969, pages 109-119.
</li>
<li>
Reuven Rubinstein,<br>
Monte Carlo Optimization, Simulation and Sensitivity of
Queueing Networks,<br>
Krieger, August 1992,<br>
ISBN: 0894647644,<br>
LC: QA298.R79.
</li>
<li>
Frederick Ruckdeschel,<br>
BASIC Scientific Subroutines,<br>
Volume I,<br>
McGraw Hill, 1980,<br>
ISBN: 0-07-054201-5,<br>
LC: QA76.95.R82.
</li>
<li>
BE Schneider,<br>
Algorithm AS 121:
Trigamma Function,<br>
Applied Statistics,<br>
Volume 27, Number 1, 1978, page 97-99.
</li>
<li>
BL Shea,<br>
Algorithm AS 239:
Chi-squared and Incomplete Gamma Integral,<br>
Applied Statistics,<br>
Volume 37, Number 3, 1988, pages 466-473.
</li>
<li>
Eric Weisstein,<br>
CRC Concise Encyclopedia of Mathematics,<br>
CRC Press, 2002,<br>
Second edition,<br>
ISBN: 1584883472,<br>
LC: QA5.W45
</li>
<li>
Michael Wichura,<br>
Algorithm AS 241:
The Percentage Points of the Normal Distribution,<br>
Applied Statistics,<br>
Volume 37, Number 3, 1988, pages 477-484.
</li>
<li>
Herbert Wilf,<br>
Some New Aspects of the Coupon Collector's Problem,<br>
SIAM Review,<br>
Volume 48, Number 3, September 2006, pages 549-565.
</li>
<li>
ML Wolfson, HV Wright,<br>
Algorithm 160:
Combinatorial of M Things Taken N at a Time,<br>
Communications of the ACM,<br>
Volume 6, Number 4, April 1963, page 161.
</li>
<li>
JC Young, Christoph Minder,<br>
Algorithm AS 76:
An Algorithm Useful in Calculating Non-Central T and
Bivariate Normal Distributions,<br>
Applied Statistics,<br>
Volume 23, Number 3, 1974, pages 455-457.
</li>
<li>
Daniel Zwillinger, Steven Kokoska,<br>
Standard Probability and Statistical Tables,<br>
CRC Press, 2000,<br>
ISBN: 1-58488-059-7,<br>
LC: QA273.3.Z95.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "prob.f90">prob.f90</a>, the source code;
</li>
<li>
<a href = "prob.sh">prob.sh</a>, commands to compile the source code;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "prob_prb.f90">prob_prb.f90</a>, the calling program;
</li>
<li>
<a href = "prob_prb.sh">prob_prb.sh</a>,
commands to compile and run the calling program;
</li>
<li>
<a href = "prob_prb_output.txt">prob_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>ANGLE_CDF</b> evaluates the Angle CDF.
</li>
<li>
<b>ANGLE_MEAN</b> returns the mean of the Angle PDF.
</li>
<li>
<b>ANGLE_PDF</b> evaluates the Angle PDF.
</li>
<li>
<b>ANGLIT_CDF</b> evaluates the Anglit CDF.
</li>
<li>
<b>ANGLIT_CDF_INV</b> inverts the Anglit CDF.
</li>
<li>
<b>ANGLIT_MEAN</b> returns the mean of the Anglit PDF.
</li>
<li>
<b>ANGLIT_PDF</b> evaluates the Anglit PDF.
</li>
<li>
<b>ANGLIT_SAMPLE</b> samples the Anglit PDF.
</li>
<li>
<b>ANGLIT_VARIANCE</b> returns the variance of the Anglit PDF.
</li>
<li>
<b>ARCSIN_CDF</b> evaluates the Arcsin CDF.
</li>
<li>
<b>ARCSIN_CDF_INV</b> inverts the Arcsin CDF.
</li>
<li>
<b>ARCSIN_CHECK</b> checks the parameter of the Arcsin CDF.
</li>
<li>
<b>ARCSIN_MEAN</b> returns the mean of the Arcsin PDF.
</li>
<li>
<b>ARCSIN_PDF</b> evaluates the Arcsin PDF.
</li>
<li>
<b>ARCSIN_SAMPLE</b> samples the Arcsin PDF.
</li>
<li>
<b>ARCSIN_VARIANCE</b> returns the variance of the Arcsin PDF.
</li>
<li>
<b>BENFORD_PDF</b> returns the Benford PDF.
</li>
<li>
<b>BESSEL_IX_VALUES</b> returns some values of the Ix Bessel function.
</li>
<li>
<b>BIRTHDAY_CDF</b> returns the Birthday Concurrence CDF.
</li>
<li>
<b>BIRTHDAY_CDF_INV</b> inverts the Birthday Concurrence CDF.
</li>
<li>
<b>BIRTHDAY_PDF</b> returns the Birthday Concurrence PDF.
</li>
<li>
<b>BERNOULLI_CDF</b> evaluates the Bernoulli CDF.
</li>
<li>
<b>BERNOULLI_CDF_INV</b> inverts the Bernoulli CDF.
</li>
<li>
<b>BERNOULLI_CHECK</b> checks the parameter of the Bernoulli CDF.
</li>
<li>
<b>BERNOULLI_MEAN</b> returns the mean of the Bernoulli PDF.
</li>
<li>
<b>BERNOULLI_PDF</b> evaluates the Bernoulli PDF.
</li>
<li>
<b>BERNOULLI_SAMPLE</b> samples the Bernoulli PDF.
</li>
<li>
<b>BERNOULLI_VARIANCE</b> returns the variance of the Bernoulli PDF.
</li>
<li>
<b>BESSEL_I0</b> evaluates the modified Bessel function I0(X).
</li>
<li>
<b>BESSEL_I0_VALUES</b> returns some values of the I0 Bessel function.
</li>
<li>
<b>BESSEL_I1</b> evaluates the Bessel I function of order I.
</li>
<li>
<b>BESSEL_I1_VALUES</b> returns some values of the I1 Bessel function.
</li>
<li>
<b>BETA</b> returns the value of the Beta function.
</li>
<li>
<b>BETA_BINOMIAL_CDF</b> evaluates the Beta Binomial CDF.
</li>
<li>
<b>BETA_BINOMIAL_CDF_INV</b> inverts the Beta Binomial CDF.
</li>
<li>
<b>BETA_BINOMIAL_CHECK</b> checks the parameters of the Beta Binomial PDF.
</li>
<li>
<b>BETA_BINOMIAL_MEAN</b> returns the mean of the Beta Binomial PDF.
</li>
<li>
<b>BETA_BINOMIAL_PDF</b> evaluates the Beta Binomial PDF.
</li>
<li>
<b>BETA_BINOMIAL_SAMPLE</b> samples the Beta Binomial CDF.
</li>
<li>
<b>BETA_BINOMIAL_VARIANCE</b> returns the variance of the Beta Binomial PDF.
</li>
<li>
<b>BETA_CDF</b> evaluates the Beta CDF.
</li>
<li>
<b>BETA_CDF_INV</b> inverts the Beta CDF.
</li>
<li>
<b>BETA_CDF_VALUES</b> returns some values of the Beta CDF.
</li>
<li>
<b>BETA_CHECK</b> checks the parameters of the Beta PDF.
</li>
<li>
<b>BETA_INC</b> returns the value of the incomplete Beta function.
</li>
<li>
<b>BETA_INC_VALUES</b> returns some values of the incomplete Beta function.
</li>
<li>
<b>BETA_MEAN</b> returns the mean of the Beta PDF.
</li>
<li>
<b>BETA_PDF</b> evaluates the Beta PDF.
</li>
<li>
<b>BETA_SAMPLE</b> samples the Beta PDF.
</li>
<li>
<b>BETA_VARIANCE</b> returns the variance of the Beta PDF.
</li>
<li>
<b>BINOMIAL_CDF</b> evaluates the Binomial CDF.
</li>
<li>
<b>BINOMIAL_CDF_VALUES</b> returns some values of the binomial CDF.
</li>
<li>
<b>BINOMIAL_CDF_INV</b> inverts the Binomial CDF.
</li>
<li>
<b>BINOMIAL_CHECK</b> checks the parameter of the Binomial PDF.
</li>
<li>
<b>BINOMIAL_COEF</b> computes the Binomial coefficient C(N,K).
</li>
<li>
<b>BINOMIAL_COEF_LOG</b> computes the logarithm of the Binomial coefficient.
</li>
<li>
<b>BINOMIAL_MEAN</b> returns the mean of the Binomial PDF.
</li>
<li>
<b>BINOMIAL_PDF</b> evaluates the Binomial PDF.
</li>
<li>
<b>BINOMIAL_SAMPLE</b> samples the Binomial PDF.
</li>
<li>
<b>BINOMIAL_VARIANCE</b> returns the variance of the Binomial PDF.
</li>
<li>
<b>BRADFORD_CDF</b> evaluates the Bradford CDF.
</li>
<li>
<b>BRADFORD_CDF_INV</b> inverts the Bradford CDF.
</li>
<li>
<b>BRADFORD_CHECK</b> checks the parameters of the Bradford PDF.
</li>
<li>
<b>BRADFORD_MEAN</b> returns the mean of the Bradford PDF.
</li>
<li>
<b>BRADFORD_PDF</b> evaluates the Bradford PDF.
</li>
<li>
<b>BRADFORD_SAMPLE</b> samples the Bradford PDF.
</li>
<li>
<b>BRADFORD_VARIANCE</b> returns the variance of the Bradford PDF.
</li>
<li>
<b>BUFFON_LAPLACE_PDF</b> evaluates the Buffon-Laplace PDF.
</li>
<li>
<b>BUFFON_LAPLACE_SIMULATE</b> simulates a Buffon-Laplace needle experiment.
</li>
<li>
<b>BUFFON_PDF</b> evaluates the Buffon PDF.
</li>
<li>
<b>BUFFON_SIMULATE</b> simulates a Buffon needle experiment.
</li>
<li>
<b>BURR_CDF</b> evaluates the Burr CDF.
</li>
<li>
<b>BURR_CDF_INV</b> inverts the Burr CDF.
</li>
<li>
<b>BURR_CHECK</b> checks the parameters of the Burr CDF.
</li>
<li>
<b>BURR_MEAN</b> returns the mean of the Burr PDF.
</li>
<li>
<b>BURR_PDF</b> evaluates the Burr PDF.
</li>
<li>
<b>BURR_SAMPLE</b> samples the Burr PDF.
</li>
<li>
<b>BURR_VARIANCE</b> returns the variance of the Burr PDF.
</li>
<li>
<b>C4_NORMAL_01_SAMPLE</b> samples the complex Normal 01 PDF.
</li>
<li>
<b>CARDIOID_CDF</b> evaluates the Cardioid CDF.
</li>
<li>
<b>CARDIOID_CDF_INV</b> inverts the Cardioid CDF.
</li>
<li>
<b>CARDIOID_CHECK</b> checks the parameters of the Cardioid CDF.
</li>
<li>
<b>CARDIOID_MEAN</b> returns the mean of the Cardioid PDF.
</li>
<li>
<b>CARDIOID_PDF</b> evaluates the Cardioid PDF.
</li>
<li>
<b>CARDIOID_SAMPLE</b> samples the Cardioid PDF.
</li>
<li>
<b>CARDIOID_VARIANCE</b> returns the variance of the Cardioid PDF.
</li>
<li>
<b>CAUCHY_CDF</b> evaluates the Cauchy CDF.
</li>
<li>
<b>CAUCHY_CDF_INV</b> inverts the Cauchy CDF.
</li>
<li>
<b>CAUCHY_CDF_VALUES</b> returns some values of the Cauchy CDF.
</li>
<li>
<b>CAUCHY_CHECK</b> checks the parameters of the Cauchy CDF.
</li>
<li>
<b>CAUCHY_MEAN</b> returns the mean of the Cauchy PDF.
</li>
<li>
<b>CAUCHY_PDF</b> evaluates the Cauchy PDF.
</li>
<li>
<b>CAUCHY_SAMPLE</b> samples the Cauchy PDF.
</li>
<li>
<b>CAUCHY_VARIANCE</b> returns the variance of the Cauchy PDF.
</li>
<li>
<b>CHI_CDF</b> evaluates the Chi CDF.
</li>
<li>
<b>CHI_CDF_INV</b> inverts the Chi CDF.
</li>
<li>
<b>CHI_CHECK</b> checks the parameters of the Chi CDF.
</li>
<li>
<b>CHI_MEAN</b> returns the mean of the Chi PDF.
</li>
<li>
<b>CHI_PDF</b> evaluates the Chi PDF.
</li>
<li>
<b>CHI_SAMPLE</b> samples the Chi PDF.
</li>
<li>
<b>CHI_VARIANCE</b> returns the variance of the Chi PDF.
</li>
<li>
<b>CHI_SQUARE_CDF</b> evaluates the Chi squared CDF.
</li>
<li>
<b>CHI_SQUARE_CDF_INV</b> inverts the Chi squared PDF.
</li>
<li>
<b>CHI_SQUARE_CDF_VALUES</b> returns some values of the Chi-Square CDF.
</li>
<li>
<b>CHI_SQUARE_CHECK</b> checks the parameter of the central Chi squared PDF.
</li>