# TRUNCATED_NORMAL The Truncated Normal Distribution

TRUNCATED_NORMAL is a FORTRAN90 library which computes quantities associated with the truncated normal distribution.

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.

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.

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.

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.

### Details

Define the unit normal distribution probability density function (PDF) for any -oo < x < +oo:

```        N(0,1)(x) = 1/sqrt(2*pi) * exp ( - x^2 / 2 )
```
This library includes the following functions for N(0,1)(x):
• normal_01_cdf(): returns CDF, given X.
• normal_01_cdf_inv(): returns X, given CDF.
• normal_01_mean(): returns the mean (which will be 0).
• normal_01_moment(): returns moments.
• normal_01_pdf(): returns PDF.
• normal_01_sample(): randomly samples.
• normal_01_variance(): returns variance (which will be 1).

For a normal distribution with mean MU and standard deviation SIGMA, the formula for the PDF is:

```        N(MU,S)(x) = 1 / sqrt(2*pi) / sigma * exp ( - ( ( x - mu ) / sigma )^2 )
```
This library includes the following functions for N(MU,SIGMA)(x):
• normal_ms_cdf(): returns CDF, given X.
• normal_ms_cdf_inv(): returns X, given CDF.
• normal_ms_mean(): returns mean (which will be MU).
• normal_ms_moment(): returns moments.
• normal_ms_moment_central(): returns central moments.
• normal_ms_pdf(): returns PDF.
• normal_ms_sample(): randomly samples.
• normal_ms_variance(): returns variance (which will be SIGMA^2).

Define the truncated normal distribution PDF with parent normal N(MU,SIGMA)(x), for a < x < b:

```        NAB(MU,SIGMA)(x) = N(MU,SIGMA)(x) / ( cdf(N(MU,SIGMA))(b) - cdf(N(MU,SIGMA))(a) )
```
This library includes the following functions for NAB(MU,SIGMA)(x)
• truncated_normal_ab_cdf(): returns CDF, given X.
• truncated_normal_ab_cdf_inv(): returns X, given CDF.
• truncated_normal_ab_mean(): returns mean.
• truncated_normal_ab_moment(): returns moments.
• truncated_normal_ab_pdf(): returns PDF.
• truncated_normal_ab_sample(): randomly samples.
• truncated_normal_ab_variance(): returns variance.

Define the lower truncated normal distribution PDF with parent normal N(MU,SIGMA)(x), for a < x < +oo:

```        NA(MU,SIGMA)(x) = N(MU,SIGMA)(x) / ( 1 - cdf(N(MU,SIGMA))(a) )
```
This library includes the following functions for NA(MU,SIGMA)(x):
• truncated_normal_a_cdf(): returns CDF, given X.
• truncated_normal_a_cdf_inv(): returns X, given CDF.
• truncated_normal_a_mean(): returns mean.
• truncated_normal_a_moment(): returns moments.
• truncated_normal_a_pdf(): returns PDF.
• truncated_normal_a_sample(): randomly samples.
• truncated_normal_a_variance(): returns variance.

Define the upper truncated normal distribution PDF with parent normal N(MU,SIGMA), for -oo < x < b:

```        NB(MU,SIGMA)(x) = N(MU,SIGMA)(x) / cdf(N(MU,SIGMA))(b)
```
This library includes the following functions for NB(MU,SIGMA)(x):
• truncated_normal_b_cdf(): returns CDF, given X.
• truncated_normal_b_cdf_inv(): returns X, given CDF.
• truncated_normal_b_mean(): returns mean.
• truncated_normal_b_moment(): returns moments.
• truncated_normal_b_pdf(): returns PDF.
• truncated_normal_b_sample(): randomly samples.
• truncated_normal_b_variance(): returns variance.

### Demonstrations

The CDF and CDF_INV functions should be inverses of each other. A simple test of the truncated AB normal functions would be

```        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 );
```
and compare x and x2, which should be quite close if the inverse function is working correctly.

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:

```        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 );
```
Typically, the values of m and ms, of v and vs should be "reasonably close".

### Languages:

TRUNCATED_NORMAL is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATHEMATICA version and a MATLAB version and a Python version..

### Related Data and Programs:

LOG_NORMAL_TRUNCATED_AB, a FORTRAN90 library which returns quantities associated with the log normal Probability Distribution Function (PDF) truncated to the interval [A,B].

NORMAL, a FORTRAN90 library which samples the normal distribution.

PDFLIB, a FORTRAN90 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.

PROB, a FORTRAN90 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.

TRUNCATED_NORMAL_RULE, a FORTRAN90 program which computes a quadrature rule for a normal distribution that has been truncated to [A,+oo), (-oo,B] or [A,B].

UNIFORM, a FORTRAN90 library which samples the uniform distribution.

### Reference:

• Norman Johnson, Samuel Kotz, Narayanaswamy Balakrishnan,
Continuous Univariate Distributions,
Second edition,
Wiley, 1994,
ISBN: 0471584940,
LC: QA273.6.J6.

### List of Routines:

• NORMAL_01_CDF evaluates the Normal 01 CDF.
• NORMAL_01_CDF_INV inverts the standard normal CDF.
• NORMAL_01_MEAN returns the mean of the Normal 01 PDF.
• NORMAL_01_MOMENT evaluates moments of the Normal 01 PDF.
• NORMAL_01_PDF evaluates the Normal 01 PDF.
• NORMAL_01_SAMPLE samples the standard normal probability distribution.
• NORMAL_01_VARIANCE returns the variance of the Normal 01 PDF.
• NORMAL_CDF evaluates the Normal CDF.
• NORMAL_CDF_INV inverts the Normal CDF.
• NORMAL_MEAN returns the mean of the Normal PDF.
• NORMAL_MOMENT evaluates moments of the Normal PDF.
• NORMAL_MOMENT_CENTRAL evaluates central moments of the Normal PDF.
• NORMAL_MOMENT_CENTRAL_VALUES: moments 0 through 10 of the Normal PDF.
• NORMAL_MOMENT_VALUES evaluates moments 0 through 8 of the Normal PDF.
• NORMAL_PDF evaluates the Normal PDF.
• NORMAL_SAMPLE samples the Normal PDF.
• NORMAL_VARIANCE returns the variance of the Normal PDF.
• R8_CHOOSE computes the binomial coefficient C(N,K) as an R8.
• R8_FACTORIAL2 computes the double factorial function.
• R8_LOG_2 returns the logarithm base 2 of an R8.
• R8_MOP returns the I-th power of -1 as an R8.
• R8_UNIFORM_01 returns a unit pseudorandom R8.
• R8POLY_VALUE evaluates an R8POLY
• R8VEC_MAX returns the maximum value in an R8VEC.
• R8VEC_MEAN returns the mean of an R8VEC.
• R8VEC_MIN returns the minimum value of an R8VEC.
• R8VEC_VARIANCE returns the variance of an R8VEC.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRUNCATED_NORMAL_AB_CDF evaluates the truncated Normal CDF.
• TRUNCATED_NORMAL_AB_CDF_VALUES: values of the Truncated Normal CDF.
• TRUNCATED_NORMAL_AB_CDF_INV inverts the truncated Normal CDF.
• TRUNCATED_NORMAL_AB_MEAN returns the mean of the truncated Normal PDF.
• TRUNCATED_NORMAL_AB_MOMENT: moments of the truncated Normal PDF.
• TRUNCATED_NORMAL_AB_PDF evaluates the truncated Normal PDF.
• TRUNCATED_NORMAL_AB_PDF_VALUES: values of the Truncated Normal PDF.
• TRUNCATED_NORMAL_AB_SAMPLE samples the truncated Normal PDF.
• TRUNCATED_NORMAL_AB_VARIANCE: variance of the truncated Normal PDF.
• TRUNCATED_NORMAL_A_CDF evaluates the lower truncated Normal CDF.
• TRUNCATED_NORMAL_A_CDF_VALUES: values of the lower Truncated Normal CDF.
• TRUNCATED_NORMAL_A_CDF_INV inverts the lower truncated Normal CDF.
• TRUNCATED_NORMAL_A_MEAN returns the mean of the lower truncated Normal PDF.
• TRUNCATED_NORMAL_A_MOMENT: moments of the lower truncated Normal PDF.
• TRUNCATED_NORMAL_A_PDF evaluates the lower truncated Normal PDF.
• TRUNCATED_NORMAL_A_PDF_VALUES: values of the lower Truncated Normal PDF.
• TRUNCATED_NORMAL_A_SAMPLE samples the lower truncated Normal PDF.
• TRUNCATED_NORMAL_A_VARIANCE: variance of the lower truncated Normal PDF.
• TRUNCATED_NORMAL_B_CDF evaluates the upper truncated Normal CDF.
• TRUNCATED_NORMAL_B_CDF_VALUES: values of the upper Truncated Normal CDF.
• TRUNCATED_NORMAL_B_CDF_INV inverts the upper truncated Normal CDF.
• TRUNCATED_NORMAL_B_MEAN returns the mean of the upper truncated Normal PDF.
• TRUNCATED_NORMAL_B_MOMENT: moments of the upper truncated Normal PDF.
• TRUNCATED_NORMAL_B_PDF evaluates the upper truncated Normal PDF.
• TRUNCATED_NORMAL_B_PDF_VALUES: values of the upper Truncated Normal PDF.
• TRUNCATED_NORMAL_B_SAMPLE samples the upper truncated Normal PDF.
• TRUNCATED_NORMAL_B_VARIANCE: variance of the upper truncated Normal PDF.

You can go up one level to the FORTRAN90 source codes.

Last revised on 17 September 2013.