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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 MATLAB 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</b> or
<b>I4_UNIFORM</b>, each of which in turn calls a routine called
<b>R8_UNIFORM_01</b>. 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 = "../../c_src/prob/prob.html">a C version</a> and
<a href = "../../cpp_src/prob/prob.html">a C++ version</a> and
<a href = "../../f77_src/prob/prob.html">a FORTRAN77 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/dcdflib/dcdflib.html">
DCDFLIB</a>,
a FORTRAN90 library which
includes routines for evaluating and inverting a number of
distributions.
</p>
<p>
<a href = "../../m_src/discrete_pdf_sample_2d/discrete_pdf_sample_2d.html">
DISCRETE_PDF_SAMPLE_2D</a>,
a MATLAB 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 = "../../m_src/normal/normal.html">
NORMAL</a>,
a MATLAB library which
samples the normal distribution.
</p>
<p>
<a href = "../../m_src/pdflib/pdflib.html">
PDFLIB</a>,
a MATLAB library which
evaluates Probability Density Functions (PDF's)
and produces random samples from them,
including beta, binomial, chi, exponential, gamma, inverse chi,
inverse gamma, multinomial, normal, scaled inverse chi, and uniform.
</p>
<p>
<a href = "../../m_src/test_values/test_values.html">
TEST_VALUES</a>,
a MATLAB library which
contains sample values for a number of distributions.
</p>
<p>
<a href = "../../m_src/truncated_normal/truncated_normal.html">
TRUNCATED_NORMAL</a>,
a MATLAB library which
works with the truncated normal distribution over [A,B], or
[A,+oo) or (-oo,B], returning the probability density function (PDF),
the cumulative density function (CDF), the inverse CDF, the mean,
the variance, and sample values.
</p>
<p>
<a href = "../../m_src/uniform/uniform.html">
UNIFORM</a>,
a MATLAB library which
samples the uniform distribution.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
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Roger Abernathy, Robert Smith,<br>
Algorithm 724,<br>
Program to Calculate F Percentiles,<br>
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</li>
<li>
Milton Abramowitz, Irene Stegun,<br>
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</li>
<li>
AG Adams,<br>
Algorithm 39:
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Computer Journal, <br>
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</li>
<li>
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Generating Gamma Variates by a Modified Rejection Technique,<br>
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</li>
<li>
Joachim Ahrens, Ulrich Dieter,<br>
Computer Methods for Sampling from Gamma, Beta, Poisson and
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Computing,<br>
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</li>
<li>
Joachim Ahrens, Klaus-Dieter Kohrt, Ulrich Dieter,<br>
Algorithm 599:
Sampling from Gamma and Poisson Distributions,<br>
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</li>
<li>
Jerry Banks, editor,<br>
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JD Beasley, SG Springer,<br>
Algorithm AS 111:
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</li>
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Frank Benford,<br>
The Law of Anomalous Numbers,<br>
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Jose Bernardo,<br>
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</li>
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</li>
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Rational Chebyshev Approximations for the Error Function,<br>
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Algorithm AS 5:
The Integral of the Non-Central T-Distribution,<br>
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Luc Devroye,<br>
Non-Uniform Random Variate Generation,<br>
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LC: QA274.D48
</li>
<li>
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</li>
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Statistical Analysis of Circular Data,<br>
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Statistical Analysis of Spherical Data,<br>
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Ted Hill,<br>
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Michael McLaughlin</a><br>
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<li>
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</li>
<li>
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</li>
<li>
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</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, CE 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 = "angle_mean.m">angle_mean.m</a>
returns the mean of the Angle PDF.
</li>
<li>
<a href = "angle_pdf.m">angle_pdf.m</a>
evaluates the Angle PDF.
</li>
<li>
<a href = "anglit_cdf.m">anglit_cdf.m</a>
evaluates the Anglit CDF.
</li>
<li>
<a href = "anglit_cdf_inv.m">anglit_cdf_inv.m</a>
inverts the Anglit CDF.
</li>
<li>
<a href = "anglit_mean.m">anglit_mean.m</a>
returns the mean of the Anglit PDF.
</li>
<li>
<a href = "anglit_pdf.m">anglit_pdf.m</a>
evaluates the Anglit PDF.
</li>
<li>
<a href = "anglit_sample.m">anglit_sample.m</a>
samples the Anglit PDF.
</li>
<li>
<a href = "anglit_variance.m">anglit_variance.m</a>
returns the variance of the Anglit PDF.
</li>
<li>
<a href = "arcsin_cdf.m">arcsin_cdf.m</a>
evaluates the Arcsin CDF.
</li>
<li>
<a href = "arcsin_check.m">arcsin_check.m</a>
checks the parameters of the Arcsin PDF.
</li>
<li>
<a href = "arcsin_cdf_inv.m">arcsin_cdf_inv.m</a>
inverts the Arcsin CDF.
</li>
<li>
<a href = "arcsin_mean.m">arcsin_mean.m</a>
returns the mean of the Arcsin PDF.
</li>
<li>
<a href = "arcsin_pdf.m">arcsin_pdf.m</a>
evaluates the Arcsin PDF.
</li>
<li>
<a href = "arcsin_sample.m">arcsin_sample.m</a>
samples the Arcsin PDF.
</li>
<li>
<a href = "arcsin_variance.m">arcsin_variance.m</a>
returns the variance of the Arcsin PDF.
</li>
<li>
<a href = "benford_pdf.m">benford_pdf.m</a>
returns the Benford probability of one or more significant digits.
</li>
<li>
<a href = "bernoulli_cdf.m">bernoulli_cdf.m</a>
evaluates the Bernoulli CDF.
</li>
<li>
<a href = "bernoulli_cdf_inv.m">bernoulli_cdf_inv.m</a>
inverts the Bernoulli CDF.
</li>
<li>
<a href = "bernoulli_check.m">bernoulli_check.m</a>
checks the parameter of the Bernoulli CDF.
</li>
<li>
<a href = "bernoulli_mean.m">bernoulli_mean.m</a>
returns the mean of the Bernoulli PDF.
</li>
<li>
<a href = "bernoulli_pdf.m">bernoulli_pdf.m</a>
evaluates the Bernoulli PDF.
</li>
<li>
<a href = "bernoulli_sample.m">bernoulli_sample.m</a>
samples the Bernoulli PDF.
</li>
<li>
<a href = "bernoulli_variance.m">bernoulli_variance.m</a>
returns the variance of the Bernoulli PDF.
</li>
<li>
<a href = "bessel_i0.m">bessel_i0.m</a>
evaluates the Bessel function I0(x).
</li>
<li>
<a href = "bessel_i0_values.m">bessel_i0_values.m</a>
returns some values of the Bessel function I0(x).
</li>
<li>
<a href = "bessel_i1.m">bessel_i1.m</a>
evaluates the Bessel function I1(x).
</li>
<li>
<a href = "bessel_i1_values.m">bessel_i1_values.m</a>
returns some values of the Bessel function I1(x).
</li>
<li>
<a href = "bessel_ix_values.m">bessel_ix_values.m</a>
returns some values of the Bessel function In(x)
for noninteger order.
</li>
<li>
<a href = "beta.m">beta.m</a>
returns the value of the Beta function.
</li>
<li>
<a href = "beta_binomial_cdf.m">beta_binomial_cdf.m</a>
evaluates the Beta Binomial CDF.
</li>
<li>
<a href = "beta_binomial_cdf_inv.m">beta_binomial_cdf_inv.m</a>
inverts the Beta Binomial CDF.
</li>
<li>
<a href = "beta_binomial_check.m">beta_binomial_check.m</a>
checks the parameters of the Beta Binomial PDF.
</li>
<li>
<a href = "beta_binomial_mean.m">beta_binomial_mean.m</a>
returns the mean of the Beta Binomial PDF.
</li>
<li>
<a href = "beta_binomial_pdf.m">beta_binomial_pdf.m</a>
evaluates the Beta Binomial PDF.
</li>
<li>
<a href = "beta_binomial_sample.m">beta_binomial_sample.m</a>
samples the Beta Binomial CDF.
</li>
<li>
<a href = "beta_binomial_variance.m">beta_binomial_variance.m</a>
returns the variance of the Beta Binomial PDF.
</li>
<li>
<a href = "beta_cdf.m">beta_cdf.m</a>
evaluates the Beta CDF.
</li>
<li>
<a href = "beta_cdf_inv.m">beta_cddf_inv.m</a>
inverts the Beta CDF.
</li>
<li>
<a href = "beta_cdf_values.m">beta_cdf_values.m</a>
returns some values of the Beta CDF.
</li>
<li>
<a href = "beta_check.m">beta_check.m</a>
checks the parameters of the Beta PDF.
</li>
<li>
<a href = "beta_inc.m">beta_inc.m</a>
returns the value of the incomplete Beta function.
</li>
<li>
<a href = "beta_inc_values.m">beta_inc_values.m</a>
returns some values of the incomplete Beta function.
</li>
<li>
<a href = "beta_mean.m">beta_mean.m</a>
returns the mean of the Beta PDF.
</li>
<li>
<a href = "beta_pdf.m">beta_pdf.m</a>
evaluates the Beta PDF.
</li>
<li>
<a href = "beta_sample.m">beta_sample.m</a>
samples the Beta PDF.
</li>
<li>
<a href = "beta_variance.m">beta_variance.m</a>
returns the variance of the Beta PDF.
</li>
<li>
<a href = "binomial_cdf.m">binomial_cdf.m</a>
evaluates the Binomial CDF.
</li>
<li>
<a href = "binomial_cdf_values.m">binomial_cdf_values.m</a>
returns some values of the binomial CDF.
</li>
<li>
<a href = "binomial_cdf_inv.m">binomial_cdf_inv.m</a>
inverts the Binomial CDF.
</li>
<li>
<a href = "binomial_cdf_values.m">binomial_cdf_values.m</a>
returns some values of the Binomial CDF.
</li>
<li>
<a href = "binomial_check.m">binomial_check.m</a>
checks the parameter of the Binomial PDF.
</li>
<li>
<a href = "binomial_coef.m">binomial_coef.m</a>
computes the Binomial coefficient C(N,K).
</li>
<li>
<a href = "binomial_coef_log.m">binomial_coef_log.m</a>
computes the logarithm of the Binomial coefficient.
</li>
<li>
<a href = "binomial_mean.m">binomial_mean.m</a>
returns the mean of the Binomial PDF.
</li>
<li>
<a href = "binomial_pdf.m">binomial_pdf.m</a>
evaluates the Binomial PDF.
</li>
<li>
<a href = "binomial_sample.m">binomial_sample.m</a>
samples the Binomial PDF.
</li>
<li>
<a href = "binomial_variance.m">binomial_variance.m</a>
returns the variance of the Binomial PDF.
</li>
<li>
<a href = "birthday_cdf.m">birthday_cdf.m</a>
evaluates the Birthday CDF.
</li>
<li>
<a href = "birthday_cdf_inv.m">birthday_cdf_inv.m</a>
inverts the Birthday CDF.
</li>
<li>
<a href = "birthday_pdf.m">birthday_pdf.m</a>
evaluates the Birthday PDF.
</li>
<li>
<a href = "bradford_cdf.m">bradford_cdf.m</a>
evaluates the Bradford CDF.
</li>
<li>
<a href = "bradford_cdf_inv.m">bradford_cdf_inv.m</a>
inverts the Bradford CDF.
</li>
<li>
<a href = "bradford_check.m">bradford_check.m</a>
checks the parameters of the Bradford PDF.
</li>
<li>
<a href = "bradford_mean.m">bradford_mean.m</a>
returns the mean of the Bradford PDF.
</li>
<li>
<a href = "bradford_pdf.m">bradford_pdf.m</a>
evaluates the Bradford PDF.
</li>
<li>
<a href = "bradford_sample.m">bradford_sample.m</a>
samples the Bradford PDF.
</li>
<li>
<a href = "bradford_variance.m">bradford_variance.m</a>
returns the variance of the Bradford PDF.
</li>
<li>
<a href = "buffon_laplace_pdf.m">buffon_laplace_pdf.m</a>
evaluates the Buffon-Laplace PDF.
</li>
<li>
<a href = "buffon_laplace_simulate.m">buffon_laplace_simulate.m</a>
simulates the Buffon-Laplace needle experiment.
</li>
<li>
<a href = "buffon_pdf.m">buffon_pdf.m</a>
evaluates the Buffon PDF.
</li>
<li>
<a href = "buffon_simulate.m">buffon_simulate.m</a>
simulates the Buffon needle experiment.
</li>
<li>
<a href = "burr_cdf.m">burr_cdf.m</a>
evaluates the Burr CDF.
</li>
<li>
<a href = "burr_cdf_inv.m">burr_cdf_inv.m</a>
inverts the Burr CDF.
</li>
<li>
<a href = "burr_check.m">burr_check.m</a>
checks the parameters of the Burr CDF.
</li>
<li>
<a href = "burr_mean.m">burr_mean.m</a>
returns the mean of the Burr PDF.
</li>
<li>
<a href = "burr_pdf.m">burr_pdf.m</a>
evaluates the Burr PDF.
</li>
<li>
<a href = "burr_sample.m">burr_sample.m</a>
samples the Burr PDF.
</li>
<li>
<a href = "burr_variance.m">burr_variance.m</a>
returns the variance of the Burr PDF.
</li>
<li>
<a href = "c_normal_01_sample.m">c_normal_01_sample.m</a>
samples the complex Normal 01 PDF.
</li>
<li>
<a href = "cardioid_cdf.m">cardioid_cdf.m</a>
evaluates the Cardioid CDF.
</li>
<li>
<a href = "cardioid_cdf_inv.m">cardioid_cdf_inv.m</a>
inverts the Cardioid CDF.
</li>
<li>
<a href = "cardioid_check.m">cardioid_check.m</a>
checks the parameters of the Cardioid CDF.
</li>
<li>
<a href = "cardioid_mean.m">cardioid_mean.m</a>
returns the mean of the Cardioid CDF.
</li>
<li>
<a href = "cardioid_pdf.m">cardioid_pdf.m</a>
evaluates the Cardioid PDF.
</li>
<li>
<a href = "cardioid_sample.m">cardioid_sample.m</a>
samples the Cardioid CDF.
</li>
<li>
<a href = "cardioid_variance.m">cardioid_variance.m</a>
returns the variance of the Cardioid CDF.
</li>
<li>
<a href = "cauchy_cdf.m">cauchy_cdf.m</a>
evaluates the Cauchy CDF.
</li>
<li>
<a href = "cauchy_cdf_inv.m">cauchy_cdf_inv.m</a>
inverts the Cauchy CDF.
</li>
<li>
<a href = "cauchy_cdf_values.m">cauchy_cdf_values.m</a>
returns some values of the Cauchy CDF.
</li>
<li>
<a href = "cauchy_check.m">cauchy_check.m</a>
checks the parameters of the Cauchy CDF.
</li>
<li>
<a href = "cauchy_mean.m">cauchy_mean.m</a>
returns the mean of the Cauchy PDF.
</li>
<li>
<a href = "cauchy_pdf.m">cauchy_pdf.m</a>
evaluates the Cauchy PDF.
</li>
<li>
<a href = "cauchy_sample.m">cauchy_sample.m</a>
samples the Cauchy PDF.
</li>
<li>
<a href = "cauchy_variance.m">cauchy_variance.m</a>
returns the variance of the Cauchy PDF.
</li>
<li>
<a href = "chi_cdf.m">chi_cdf.m</a>
evaluates the Chi CDF.
</li>
<li>
<a href = "chi_cdf_inv.m">chi_cdf_inv.m</a>
inverts the Chi CDF.
</li>
<li>
<a href = "chi_check.m">chi_check.m</a>
checks the parameters of the Chi CDF.
</li>
<li>
<a href = "chi_mean.m">chi_mean.m</a>
returns the mean of the Chi PDF.
</li>
<li>
<a href = "chi_pdf.m">chi_pdf.m</a>
evaluates the Chi PDF.
</li>
<li>
<a href = "chi_sample.m">chi_sample.m</a>
samples the Chi PDF.
</li>
<li>
<a href = "chi_variance.m">chi_variance.m</a>
returns the variance of the Chi PDF.
</li>
<li>
<a href = "chi_square_cdf.m">chi_square_cdf.m</a>
evaluates the chi-squared CDF.
</li>
<li>
<a href = "chi_square_cdf_inv.m">chi_square_cdf_inv.m</a>
inverts the chi-squared PDF.
</li>
<li>
<a href = "chi_square_cdf_values.m">chi_square_cdf_values.m</a>
returns values of the Chi-Square central CDF.
</li>
<li>
<a href = "chi_square_check.m">chi_square_check.m</a>
checks the parameter of the central Chi-Squared PDF.
</li>
<li>
<a href = "chi_square_mean.m">chi_square_mean.m</a>
returns the mean of the central Chi-Squared PDF.
</li>
<li>
<a href = "chi_square_pdf.m">chi_square_pdf.m</a>
evaluates the central Chi-Squared PDF.
</li>
<li>
<a href = "chi_square_sample.m">chi_square_sample.m</a>
samples the central Chisquare PDF.
</li>
<li>
<a href = "chi_square_variance.m">chi_square_variance.m</a>
returns the variance of the central Chisquare PDF.
</li>
<li>
<a href = "chi_square_noncentral_check.m">
chi_square_noncentral_check.m</a>
checks the parameters of the noncentral Chi-Squared PDF.
</li>
<li>
<a href = "chi_square_noncentral_cdf_values.m">
chi_square_noncentral_cdf_values.m</a>
returns some values of the noncentral Chi-Squared CDF.
</li>
<li>
<a href = "chi_square_noncentral_mean.m">
chi_square_noncentral_mean.m</a>
returns the mean of the noncentral Chi-Squared PDF.