stdlib_stats_distribution_normal.md

title
stats_distribution_normal

Statistical Distributions -- Normal Distribution Module

[TOC]

rvs_normal - normal distribution random variates

Status

Experimental

Description

A normal continuous random variate distribution, also known as Gaussian, or Gauss or Laplace-Gauss distribution. The location loc specifies the mean or expectation ((\mu)). The scale specifies the standard deviation ((\sigma)).

Without argument, the function returns a standard normal distributed random variate (N(0,1)).

With two arguments, the function returns a normal distributed random variate (N(\mu=\text{loc}, \sigma^2=\text{scale}^2)). For complex arguments, the real and imaginary parts are independent of each other.

With three arguments, the function returns a rank-1 array of normal distributed random variates.

@note The algorithm used for generating exponential random variates is fundamentally limited to double precision.¹

Syntax

result = [[stdlib_stats_distribution_normal(module):rvs_normal(interface)]]([loc, scale] [[, array_size]])

Class

Elemental function (passing both loc and scale).

Arguments

loc: optional argument has intent(in) and is a scalar of type real or complex.

scale: optional argument has intent(in) and is a positive scalar of type real or complex.

array_size: optional argument has intent(in) and is a scalar of type integer.

loc and scale arguments must be of the same type.

Return value

The result is a scalar or rank-1 array, with a size of array_size, and the same type as scale and loc. If scale is non-positive, the result is NaN.

Example

{!example/stats_distribution_normal/example_normal_rvs.f90!}

pdf_normal - normal distribution probability density function

Status

Experimental

Description

The probability density function (pdf) of the single real variable normal distribution:

$$f(x) = \frac{1}{\sigma \sqrt{2}} \exp{\left[-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right]}$$

For a complex varible ( z=(x + y i) ) with independent real ( x ) and imaginary ( y ) parts, the joint probability density function is the product of the the corresponding real and imaginary marginal pdfs:²

$$f(x + y \mathit{i}) = f(x) f(y) = \frac{1}{2\sigma_{x}\sigma_{y}} \exp{\left[-\frac{1}{2}\left(\left(\frac{x-\mu_x}{\sigma_{x}}\right)^{2}+\left(\frac{y-\mu_y}{\sigma_{y}}\right)^{2}\right)\right]}$$

Syntax

result = [[stdlib_stats_distribution_normal(module):pdf_normal(interface)]](x, loc, scale)

Class

Elemental function

Arguments

x: has intent(in) and is a scalar of type real or complex.

loc: has intent(in) and is a scalar of type real or complex.

scale: has intent(in) and is a positive scalar of type real or complex.

All three arguments must have the same type.

Return value

The result is a scalar or an array, with a shape conformable to the arguments, and the same type as the input arguments. If scale is non-positive, the result is NaN.

Example

{!example/stats_distribution_normal/example_normal_pdf.f90!}

cdf_normal - normal distribution cumulative distribution function

Status

Experimental

Description

Cumulative distribution function of the single real variable normal distribution:

$$F(x) = \frac{1}{2}\left [ 1+\text{erf}\left(\frac{x-\mu}{\sigma \sqrt{2}}\right) \right ]$$

For the complex variable ( z=(x + y i) ) with independent real ( x ) and imaginary ( y ) parts, the joint cumulative distribution function is the product of the corresponding real and imaginary marginal cdfs:²

$$ F(x+y\mathit{i})=F(x)F(y)=\frac{1}{4} \left[ 1+\text{erf}\left(\frac{x-\mu_x}{\sigma_x \sqrt{2}}\right) \right] \left[ 1+\text{erf}\left(\frac{y-\mu_y}{\sigma_y \sqrt{2}}\right) \right] $$

Syntax

result = [[stdlib_stats_distribution_normal(module):cdf_normal(interface)]](x, loc, scale)

Class

Elemental function

Arguments

x: has intent(in) and is a scalar of type real or complex.

loc: has intent(in) and is a scalar of type real or complex.

scale: has intent(in) and is a positive scalar of type real or complex.

All three arguments must have the same type.

Return value

The result is a scalar or an array, with a shape conformable to the arguments, and the same type as the input arguments. If scale is non-positive, the result is NaN.

Example

{!example/stats_distribution_normal/example_normal_cdf.f90!}

Footnotes

1. Marsaglia, George, and Wai Wan Tsang. "The ziggurat method for generating random variables." Journal of statistical software 5 (2000): 1-7. ↩

2. Miller, Scott, and Donald Childers. Probability and random processes: With applications to signal processing and communications. Academic Press, 2012 (p. 197). ↩ ↩²