PROB
Probability Density Functions
PROB,
a C++ library which
handles various discrete and
continuous probability density functions (PDF's).
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.
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.
In some cases, the inverse of the CDF can easily be computed.
If
X = CDF_INV ( P )
then we are asserting that the value X has a cumulative
probability density function of P, in other words,
the probability that the variable is less than or equal to X
is P. 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.
For most distributions, the mean 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.
For most distributions, the variance 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 standard
deviation. For the distributions covered here, the variances are
often known beforehand, and no summation or integration is required.
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 R8_RANDOM or
I4_RANDOM, each of which in turn calls a routine called
R8_UNIFORM_01. You may prefer a different random number generator
for this purpose.
Licensing:
The computer code and data files described and made available on this web page
are distributed under
the GNU LGPL license.
Languages:
PROB is available in
a C version and
a C++ version and
a FORTRAN90 version and
a MATLAB version and
a Python version.
Related Data and Programs:
ASA152,
a C++ library which
evaluates point and cumulative probabilities associated with the
hypergeometric distribution;
this is Applied Statistics Algorithm 152;
ASA226,
a C++ library which
evaluates the CDF of the noncentral Beta distribution.
ASA241,
a C++ library which
evaluates the percentage points of the normal distribution.
ASA243,
a C++ library which
evaluates the CDF of the noncentral T distribution.
ASA310,
a C++ library which
computes the CDF of the noncentral Beta distribution.
BETA_NC,
a C++ library which
evaluates the CDF of the noncentral Beta distribution.
CDFLIB,
a C++ library which
evaluates the cumulative density function (CDF), inverse CDF,
and certain other inverse functions, for distributions including
beta, binomial, chisquare, noncentral chisquare, F, noncentral F,
gamma, negative binomial, normal, Poisson, and students T,
by Barry Brown, James Lovato, Kathy Russell.
DISCRETE_PDF_SAMPLE_2D,
a C++ 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.
gsl_test,
a C++ library which
includes many routines for evaluating probability distributions.
LOG_NORMAL,
a C++ library which
returns quantities associated with the log normal Probability
Distribution Function (PDF).
LOG_NORMAL_TRUNCATED_AB,
a C++ library which
returns quantities associated with the log normal Probability
Distribution Function (PDF) truncated to the interval [A,B].
NORMAL,
a C++ library which
samples the normal distribution.
prob_test
RANDOM_DATA,
a C++ library which
generates sample points for
various probability distributions, spatial dimensions, and geometries;
TEST_VALUES,
a C++ library which
contains sample values for a number of distributions.
TRUNCATED_NORMAL,
a C++ 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.
UNIFORM,
a C++ library which
samples the uniform distribution.
ZIGGURAT,
a C++ program which
generates points from a uniform, normal or exponential distribution, using
the ziggurat method.
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Source Code:
Last revised on 01 April 2020.