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<html>
<head>
<title>
TRUNCATED_NORMAL - The Truncated Normal Distribution
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRUNCATED_NORMAL <br> The Truncated Normal Distribution
</h1>
<hr>
<p>
<b>TRUNCATED_NORMAL</b>
is a C++ library which
computes quantities associated with the truncated normal distribution.
</p>
<p>
In statistics and probability, many quantities are well modeled by the
normal distribution, often called the "bell curve". The main
features of the normal distribution are that it has an average value
or mean, whose probability exceeds that of all other values, and that on
either side of the mean, the density function smoothly decreases, without
every becoming zero.
</p>
<p>
For various reasons, it may be preferable to work with a truncated normal
distribution. This may be because the normal distribution is a good fit
for our data, but for physical reasons we know our data can never be
negative, or we only wish to consider data within a particular range of
interest to us, which we might symbolize as [A,B], or [A,+oo), or (-oo,B),
depending on the truncation we apply.
</p>
<p>
It is possible to define a truncated normal distribution by first assuming
the existence of a "parent" normal distribution, with mean MU and standard
deviation SIGMA. We may then derive a modified distribution which is zero
outside the region of interest, and inside the region, has the same
"shape" as the parent normal distribution, although scaled by a constant
so that its integral is 1.
</p>
<p>
Note that, although we define the truncated normal distribution function
in terms of a parent normal distribution with mean MU and standard
deviation SIGMA, in general, the mean and standard deviation of the
truncated normal distribution are different values entirely; however,
their values can be worked out from the parent values MU and SIGMA, and
the truncation limits. That is what is done in the "_mean()" and
"_variance()" functions.
</p>
<h3 align = "center">
Details
</h3>
<p>
Define the unit normal distribution probability density function
(PDF) for any -oo < x < +oo:
<pre>
N(0,1)(x) = 1/sqrt(2*pi) * exp ( - x^2 / 2 )
</pre>
This library includes the following functions for N(0,1)(x):
<ul>
<li>
normal_01_cdf(): returns CDF, given X.
</li>
<li>
normal_01_cdf_inv(): returns X, given CDF.
</li>
<li>
normal_01_mean(): returns the mean (which will be 0).
</li>
<li>
normal_01_moment(): returns moments.
</li>
<li>
normal_01_pdf(): returns PDF.
</li>
<li>
normal_01_sample(): randomly samples.
</li>
<li>
normal_01_variance(): returns variance (which will be 1).
</li>
</ul>
</p>
<p>
For a normal distribution with mean MU and standard deviation SIGMA,
the formula for the PDF is:
<pre>
N(MU,S)(x) = 1 / sqrt(2*pi) / sigma * exp ( - ( ( x - mu ) / sigma )^2 )
</pre>
This library includes the following functions for N(MU,SIGMA)(x):
<ul>
<li>
normal_ms_cdf(): returns CDF, given X.
</li>
<li>
normal_ms_cdf_inv(): returns X, given CDF.
</li>
<li>
normal_ms_mean(): returns mean (which will be MU).
</li>
<li>
normal_ms_moment(): returns moments.
</li>
<li>
normal_ms_moment_central(): returns central moments.
</li>
<li>
normal_ms_pdf(): returns PDF.
</li>
<li>
normal_ms_sample(): randomly samples.
</li>
<li>
normal_ms_variance(): returns variance (which will be SIGMA^2).
</li>
</ul>
</p>
<p>
Define the truncated normal distribution PDF
with parent normal N(MU,SIGMA)(x), for a < x < b:
<pre>
NAB(MU,SIGMA)(x) = N(MU,SIGMA)(x) / ( cdf(N(MU,SIGMA))(b) - cdf(N(MU,SIGMA))(a) )
</pre>
This library includes the following functions for NAB(MU,SIGMA)(x)
<ul>
<li>
truncated_normal_ab_cdf(): returns CDF, given X.
</li>
<li>
truncated_normal_ab_cdf_inv(): returns X, given CDF.
</li>
<li>
truncated_normal_ab_mean(): returns mean.
</li>
<li>
truncated_normal_ab_moment(): returns moments.
</li>
<li>
truncated_normal_ab_pdf(): returns PDF.
</li>
<li>
truncated_normal_ab_sample(): randomly samples.
</li>
<li>
truncated_normal_ab_variance(): returns variance.
</li>
</ul>
</p>
<p>
Define the lower truncated normal distribution PDF
with parent normal N(MU,SIGMA)(x), for a < x < +oo:
<pre>
NA(MU,SIGMA)(x) = N(MU,SIGMA)(x) / ( 1 - cdf(N(MU,SIGMA))(a) )
</pre>
This library includes the following functions for NA(MU,SIGMA)(x):
<ul>
<li>
truncated_normal_a_cdf(): returns CDF, given X.
</li>
<li>
truncated_normal_a_cdf_inv(): returns X, given CDF.
</li>
<li>
truncated_normal_a_mean(): returns mean.
</li>
<li>
truncated_normal_a_moment(): returns moments.
</li>
<li>
truncated_normal_a_pdf(): returns PDF.
</li>
<li>
truncated_normal_a_sample(): randomly samples.
</li>
<li>
truncated_normal_a_variance(): returns variance.
</li>
</ul>
</p>
<p>
Define the upper truncated normal distribution PDF
with parent normal N(MU,SIGMA), for -oo < x < b:
<pre>
NB(MU,SIGMA)(x) = N(MU,SIGMA)(x) / cdf(N(MU,SIGMA))(b)
</pre>
This library includes the following functions for NB(MU,SIGMA)(x):
<ul>
<li>
truncated_normal_b_cdf(): returns CDF, given X.
</li>
<li>
truncated_normal_b_cdf_inv(): returns X, given CDF.
</li>
<li>
truncated_normal_b_mean(): returns mean.
</li>
<li>
truncated_normal_b_moment(): returns moments.
</li>
<li>
truncated_normal_b_pdf(): returns PDF.
</li>
<li>
truncated_normal_b_sample(): randomly samples.
</li>
<li>
truncated_normal_b_variance(): returns variance.
</li>
</ul>
</p>
<h3 align = "center">
Demonstrations
</h3>
<p>
The CDF and CDF_INV functions should be inverses
of each other. A simple test of the truncated AB normal
functions would be
<pre>
mu = 100.0;
sigma = 25.0;
a = 50.0;
b = 120.0;
seed = 123456789;
[ x, seed ] = truncated_normal_ab_sample ( mu, sigma, a, b, seed );
cdf = truncated_normal_ab_cdf ( x, mu, sigma, a, b );
x2 = truncated_normal_ab_cdf_inv ( cdf, mu, sigma, a, b );
</pre>
and compare <b>x</b> and <b>x2</b>, which should be quite close if the
inverse function is working correctly.
</p>
<p>
A simple test of the mean and variance functions might be to compare the
theoretical mean and variance to the sample mean and variance of a sample
of 1,000 values:
<pre>
sample_num = 1000;
mu = 100.0;
sigma = 25.0;
a = 50.0;
b = 120.0;
seed = 123456789;
for i = 1 : sample_num
[ x(i), seed ] = truncated_normal_ab_sample ( mu, sigma, a, b, seed );
end
m = truncated_normal_ab_mean ( mu, sigma, a, b );
v = truncated_normal_ab_variance ( mu, sigma, a, b );
ms = mean ( x );
vs = var ( x );
</pre>
Typically, the values of <b>m</b> and <b>ms</b>, of <b>v</b> and <b>vs</b>
should be "reasonably close".
</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>TRUNCATED_NORMAL</b> is available in
<a href = "../../c_src/truncated_normal/truncated_normal.html">a C version</a> and
<a href = "../../cpp_src/truncated_normal/truncated_normal.html">a C++ version</a> and
<a href = "../../f77_src/truncated_normal/truncated_normal.html">a FORTRAN77 version</a> and
<a href = "../../f_src/truncated_normal/truncated_normal.html">a FORTRAN90 version</a> and
<a href = "../../math_src/truncated_normal/truncated_normal.html">a MATHEMATICA version</a> and
<a href = "../../m_src/truncated_normal/truncated_normal.html">a MATLAB version</a> and
<a href = "../../py_src/truncated_normal/truncated_normal.html">a Python version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/normal/normal.html">
NORMAL</a>,
a C++ library which
samples the normal distribution.
</p>
<p>
<a href = "../../cpp_src/pdflib/pdflib.html">
PDFLIB</a>,
a C++ 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 = "../../cpp_src/prob/prob.html">
PROB</a>,
a C++ library which
evaluates Probability Density Functions (PDF's) and
Cumulative Density Functions (CDF's), means, variances, and
samples for a variety of standard probability distributions.
</p>
<p>
<a href = "../../cpp_src/truncated_normal_rule/truncated_normal_rule.html">
TRUNCATED_NORMAL_RULE</a>,
a C++ program which
computes a quadrature rule for a normal probability density function (PDF),
also called a Gaussian distribution, that has been
truncated to [A,+oo), (-oo,B] or [A,B].
</p>
<p>
<a href = "../../cpp_src/uniform/uniform.html">
UNIFORM</a>,
a C++ library which
samples the uniform distribution.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ul>
<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>
</ul>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "truncated_normal.cpp">truncated_normal.cpp</a>, the source code;
</li>
<li>
<a href = "truncated_normal.hpp">truncated_normal.hpp</a>, the include file;
</li>
<li>
<a href = "truncated_normal.sh">truncated_normal.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "truncated_normal_prb.cpp">truncated_normal_prb.cpp</a>, the calling program;
</li>
<li>
<a href = "truncated_normal_prb.sh">truncated_normal_prb.sh</a>,
commands to compile, link and run the calling program;
</li>
<li>
<a href = "truncated_normal_prb_output.txt">truncated_normal_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>NORMAL_01_CDF</b> evaluates the Normal 01 CDF.
</li>
<li>
<b>NORMAL_01_CDF_INV</b> inverts the standard normal CDF.
</li>
<li>
<b>NORMAL_01_MEAN</b> returns the mean of the Normal 01 PDF.
</li>
<li>
<b>NORMAL_01_MOMENT</b> evaluates moments of the Normal 01 PDF.
</li>
<li>
<b>NORMAL_01_PDF</b> evaluates the Normal 01 PDF.
</li>
<li>
<b>NORMAL_01_SAMPLE</b> samples the standard normal probability distribution.
</li>
<li>
<b>NORMAL_01_VARIANCE</b> returns the variance of the Normal 01 PDF.
</li>
<li>
<b>NORMAL_CDF</b> evaluates the Normal CDF.
</li>
<li>
<b>NORMAL_CDF_INV</b> inverts the Normal CDF.
</li>
<li>
<b>NORMAL_MEAN</b> returns the mean of the Normal PDF.
</li>
<li>
<b>NORMAL_MOMENT</b> evaluates moments of the Normal PDF.
</li>
<li>
<b>NORMAL_MOMENT_CENTRAL</b> evaluates central moments of the Normal PDF.
</li>
<li>
<b>NORMAL_MOMENT_CENTRAL_VALUES:</b> moments 0 through 10 of the Normal PDF.
</li>
<li>
<b>NORMAL_MOMENT_VALUES</b> evaluates moments 0 through 8 of the Normal PDF.
</li>
<li>
<b>NORMAL_PDF</b> evaluates the Normal PDF.
</li>
<li>
<b>NORMAL_SAMPLE</b> samples the Normal PDF.
</li>
<li>
<b>NORMAL_VARIANCE</b> returns the variance of the Normal PDF.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_CHOOSE</b> computes the binomial coefficient C(N,K) as an R8.
</li>
<li>
<b>R8_FACTORIAL2</b> computes the double factorial function.
</li>
<li>
<b>R8_HUGE</b> returns a "huge" R8.
</li>
<li>
<b>R8_LOG_2</b> returns the logarithm base 2 of the absolute value of an R8.
</li>
<li>
<b>R8_MOP</b> returns the I-th power of -1 as an R8 value.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</li>
<li>
<b>R8POLY_VALUE</b> evaluates a double precision polynomial.
</li>
<li>
<b>R8VEC_MAX</b> returns the value of the maximum element in an R8VEC.
</li>
<li>
<b>R8VEC_MEAN</b> returns the mean of an R8VEC.
</li>
<li>
<b>R8VEC_MIN</b> returns the value of the minimum element in an R8VEC.
</li>
<li>
<b>R8VEC_VARIANCE</b> returns the variance of an R8VEC.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TRUNCATED_NORMAL_AB_CDF</b> evaluates the truncated Normal CDF.
</li>
<li>
<b>TRUNCATED_NORMAL_AB_CDF_VALUES:</b> values of the Truncated Normal CDF.
</li>
<li>
<b>TRUNCATED_NORMAL_AB_CDF_INV</b> inverts the truncated Normal CDF.
</li>
<li>
<b>TRUNCATED_NORMAL_AB_MEAN</b> returns the mean of the truncated Normal PDF.
</li>
<li>
<b>TRUNCATED_NORMAL_AB_MOMENT:</b> moments of the truncated Normal PDF.
</li>
<li>
<b>TRUNCATED_NORMAL_AB_PDF</b> evaluates the truncated Normal PDF.
</li>
<li>
<b>TRUNCATED_NORMAL_AB_PDF_VALUES:</b> values of the Truncated Normal PDF.
</li>
<li>
<b>TRUNCATED_NORMAL_AB_SAMPLE</b> samples the truncated Normal PDF.
</li>
<li>
<b>TRUNCATED_NORMAL_AB_VARIANCE</b> returns the variance of the truncated Normal PDF.
</li>
<li>
<b>TRUNCATED_NORMAL_A_CDF</b> evaluates the lower truncated Normal CDF.
</li>
<li>
<b>TRUNCATED_NORMAL_A_CDF_VALUES:</b> values of the Lower Truncated Normal CDF.
</li>
<li>
<b>TRUNCATED_NORMAL_A_CDF_INV</b> inverts the lower truncated Normal CDF.
</li>
<li>
<b>TRUNCATED_NORMAL_A_MEAN</b> returns the mean of the lower truncated Normal PDF.
</li>
<li>
<b>TRUNCATED_NORMAL_A_MOMENT:</b> moments of the lower truncated Normal PDF.
</li>
<li>
<b>TRUNCATED_NORMAL_A_PDF</b> evaluates the lower truncated Normal PDF.
</li>
<li>
<b>TRUNCATED_NORMAL_A_PDF_VALUES:</b> values of the Lower Truncated Normal PDF.
</li>
<li>
<b>TRUNCATED_NORMAL_A_SAMPLE</b> samples the lower truncated Normal PDF.
</li>
<li>
<b>TRUNCATED_NORMAL_A_VARIANCE:</b> variance of the lower truncated Normal PDF.
</li>
<li>
<b>TRUNCATED_NORMAL_B_CDF</b> evaluates the upper truncated Normal CDF.
</li>
<li>
<b>TRUNCATED_NORMAL_B_CDF_VALUES:</b> values of the upper Truncated Normal CDF.
</li>
<li>
<b>TRUNCATED_NORMAL_B_CDF_INV</b> inverts the upper truncated Normal CDF.
</li>
<li>
<b>TRUNCATED_NORMAL_B_MEAN</b> returns the mean of the upper truncated Normal PDF.
</li>
<li>
<b>TRUNCATED_NORMAL_B_MOMENT:</b> moments of the upper truncated Normal PDF.
</li>
<li>
<b>TRUNCATED_NORMAL_B_PDF</b> evaluates the upper truncated Normal PDF.
</li>
<li>
<b>TRUNCATED_NORMAL_B_PDF_VALUES:</b> values of the Upper Truncated Normal PDF.
</li>
<li>
<b>TRUNCATED_NORMAL_B_SAMPLE</b> samples the upper truncated Normal PDF.
</li>
<li>
<b>TRUNCATED_NORMAL_B_VARIANCE:</b> variance of the upper truncated Normal PDF.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 11 September 2013.
</i>
<!-- John Burkardt -->
</body>
</html>