Normal Distribution¶

Table of contents

Density Function ¶

The density function of the Normal (Gaussian) distribution:

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

Methods for scalar input, as well as for vector/matrix input, are listed below.

Scalar Input ¶

template<typename T1, typename T2, typename T3> constexpr common_return_t<T1, T2, T3> dnorm(const T1 x, const T2 mu_par, const T3 sigma_par, const bool log_form = false) noexcept¶

Density function of the Normal distribution.

Example:

stats::dnorm(0.5,1.0,2.0,false); 

Parameters

x – a real-valued input.
mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.
log_form – return the log-density or the true form.

Returns

the density function evaluated at x.

template<typename T> constexpr return_t<T> dnorm(const T x, const bool log_form = false) noexcept¶

Density function of the standard Normal distribution.

Example:

stats::dnorm(0.5,false); 

Parameters

x – a real-valued input.
log_form – return the log-density or the true form.

Returns

the density function evaluated at x.

template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>> inline std::vector<rT> dnorm(const std::vector<eT> &x, const T1 mu_par, const T2 sigma_par, const bool log_form = false)¶

Density function of the Normal distribution.

Example:

std::vector<double> x = {0.0, 1.0, 2.0};
stats::dnorm(x,1.0,2.0,false);

Parameters

x – a standard vector.
mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.
log_form – return the log-density or the true form.

Returns

a vector of density function values corresponding to the elements of x.

Armadillo ¶

template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>> inline ArmaMat<rT> dnorm(const ArmaMat<eT> &X, const T1 mu_par, const T2 sigma_par, const bool log_form = false)¶

Density function of the Normal distribution.

Example:

arma::mat X = { {0.2, -1.7,  0.1},
                {0.9,  4.0, -0.3} };
stats::dnorm(X,1.0,1.0,false);

Parameters

X – a matrix of input values.
mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.
log_form – return the log-density or the true form.

Returns

a matrix of density function values corresponding to the elements of X.

Blaze ¶

template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>, bool To = blaze::columnMajor> inline BlazeMat<rT, To> dnorm(const BlazeMat<eT, To> &X, const T1 mu_par, const T2 sigma_par, const bool log_form = false)¶

Density function of the Normal distribution.

Example:

stats::dnorm(X,1.0,1.0,false);

Parameters

X – a matrix of input values.
mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.
log_form – return the log-density or the true form.

Returns

a matrix of density function values corresponding to the elements of X.

Eigen ¶

template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>, int iTr = Eigen::Dynamic, int iTc = Eigen::Dynamic> inline EigenMat<rT, iTr, iTc> dnorm(const EigenMat<eT, iTr, iTc> &X, const T1 mu_par, const T2 sigma_par, const bool log_form = false)¶

Density function of the Normal distribution.

Example:

stats::dnorm(X,1.0,1.0,false);

Parameters

X – a matrix of input values.
mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.
log_form – return the log-density or the true form.

Returns

a matrix of density function values corresponding to the elements of X.

Cumulative Distribution Function ¶

The cumulative distribution function of the Normal (Gaussian) distribution:

\[F(x; \mu, \sigma) = \int_{-\infty}^x f(z; \mu, \sigma) dz = \frac{1}{2} \times \left( 1 + \text{erf} \left( \frac{x - \mu}{\sqrt{2} \sigma} \right) \right)\]

where \(\text{erf}(\cdot)\) denotes the Gaussian error function.

Methods for scalar input, as well as for vector/matrix input, are listed below.

Scalar Input ¶

template<typename T1, typename T2, typename T3> constexpr common_return_t<T1, T2, T3> pnorm(const T1 x, const T2 mu_par, const T3 sigma_par, const bool log_form = false) noexcept¶

Distribution function of the Normal distribution.

Example:

stats::pnorm(2.0,1.0,2.0,false); 

Parameters

x – a real-valued input.
mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.
log_form – return the log-probability or the true form.

Returns

the cumulative distribution function evaluated at x.

template<typename T> constexpr return_t<T> pnorm(const T x, const bool log_form = false) noexcept¶

Distribution function of the standard Normal distribution.

Example:

stats::pnorm(1.0,false); 

Parameters

x – a real-valued input.
log_form – return the log-probability or the true form.

Returns

the cumulative distribution function evaluated at x.

Vector/Matrix Input ¶

STL Containers ¶

template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>> inline std::vector<rT> pnorm(const std::vector<eT> &x, const T1 mu_par, const T2 sigma_par, const bool log_form = false)¶

Distribution function of the Normal distribution.

Example:

std::vector<double> x = {0.0, 1.0, 2.0};
stats::pnorm(x,1.0,2.0,false);

Parameters

x – a standard vector.
mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.
log_form – return the log-probability or the true form.

Returns

a vector of CDF values corresponding to the elements of x.

Armadillo ¶

template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>> inline ArmaMat<rT> pnorm(const ArmaMat<eT> &X, const T1 mu_par, const T2 sigma_par, const bool log_form = false)¶

Distribution function of the Normal distribution.

Example:

arma::mat X = { {0.2, -1.7,  0.1},
                {0.9,  4.0, -0.3} };
stats::pnorm(X,1.0,1.0,false);

Parameters

X – a matrix of input values.
mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.
log_form – return the log-probability or the true form.

Returns

a matrix of CDF values corresponding to the elements of X.

Blaze ¶

template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>, bool To = blaze::columnMajor> inline BlazeMat<rT, To> pnorm(const BlazeMat<eT, To> &X, const T1 mu_par, const T2 sigma_par, const bool log_form = false)¶

Distribution function of the Normal distribution.

Example:

stats::pnorm(X,1.0,1.0,false);

Parameters

X – a matrix of input values.
mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.
log_form – return the log-probability or the true form.

Returns

a matrix of CDF values corresponding to the elements of X.

Eigen ¶

template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>, int iTr = Eigen::Dynamic, int iTc = Eigen::Dynamic> inline EigenMat<rT, iTr, iTc> pnorm(const EigenMat<eT, iTr, iTc> &X, const T1 mu_par, const T2 sigma_par, const bool log_form = false)¶

Distribution function of the Normal distribution.

Example:

stats::pnorm(X,1.0,1.0,false);

Parameters

X – a matrix of input values.
mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.
log_form – return the log-probability or the true form.

Returns

a matrix of CDF values corresponding to the elements of X.

Quantile Function ¶

The quantile function of the log-Normal distribution:

\[q(p; \mu, \sigma) = \mu + \sqrt{2} \sigma \times \text{erf}^{-1} \left( 2 p - 1 \right)\]

where \(\text{erf}^{-1}(\cdot)\) denotes the inverse Gaussian error function.

Methods for scalar input, as well as for vector/matrix input, are listed below.

Scalar Input ¶

template<typename T1, typename T2, typename T3> constexpr common_return_t<T1, T2, T3> qnorm(const T1 p, const T2 mu_par, const T3 sigma_par) noexcept¶

Quantile function of the Normal distribution.

Example:

stats::qnorm(0.5,1.0,2.0); 

Parameters

p – a real-valued input.
mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.

Returns

the quantile function evaluated at p.

template<typename T> constexpr return_t<T> qnorm(const T p) noexcept¶

Quantile function of the standard Normal distribution.

Example:

stats::qnorm(0.5); 

Parameters: p – a real-valued input.
Returns: the quantile function evaluated at p.

Vector/Matrix Input ¶

STL Containers ¶

template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>> inline std::vector<rT> qnorm(const std::vector<eT> &x, const T1 mu_par, const T2 sigma_par)¶

Quantile function of the Normal distribution.

Example:

std::vector<double> x = {0.1, 0.3, 0.7};
stats::qnorm(x,1.0,2.0);

Parameters

x – a standard vector.
mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.

Returns

a vector of quantile values corresponding to the elements of x.

Armadillo ¶

template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>> inline ArmaMat<rT> qnorm(const ArmaMat<eT> &X, const T1 mu_par, const T2 sigma_par)¶

Quantile function of the Normal distribution.

Example:

arma::mat X = { {0.2, 0.7, 0.9},
                {0.1, 0.8, 0.3} };
stats::qnorm(X,1.0,1.0);

Parameters

X – a matrix of input values.
mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.

Returns

a matrix of quantile values corresponding to the elements of X.

Blaze ¶

template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>, bool To = blaze::columnMajor> inline BlazeMat<rT, To> qnorm(const BlazeMat<eT, To> &X, const T1 mu_par, const T2 sigma_par)¶

Quantile function of the Normal distribution.

Example:

stats::qnorm(X,1.0,1.0);

Parameters

X – a matrix of input values.
mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.

Returns

a matrix of quantile values corresponding to the elements of X.

Eigen ¶

template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>, int iTr = Eigen::Dynamic, int iTc = Eigen::Dynamic> inline EigenMat<rT, iTr, iTc> qnorm(const EigenMat<eT, iTr, iTc> &X, const T1 mu_par, const T2 sigma_par)¶

Quantile function of the Normal distribution.

Example:

stats::qnorm(X,1.0,1.0);

Parameters

X – a matrix of input values.
mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.

Returns

a matrix of quantile values corresponding to the elements of X.

Random Sampling ¶

Random sampling for the Normal distribution is achieved via the normal_distribution class from the C++ standard library, imported from <random>.

Scalar Output ¶

Random number engines

template<typename T1, typename T2> inline common_return_t<T1, T2> rnorm(const T1 mu_par, const T2 sigma_par, rand_engine_t &engine)¶

Random sampling function for the Normal distribution.

Example:

stats::rand_engine_t engine(1776);
stats::rnorm(1.0,2.0,engine);

Parameters

mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.
engine – a random engine, passed by reference.

Returns

a pseudo-random draw from the Normal distribution.

Seed values

template<typename T1, typename T2> inline common_return_t<T1, T2> rnorm(const T1 mu_par, const T2 sigma_par, const ullint_t seed_val = std::random_device{}())¶

Random sampling function for the Normal distribution.

Example:

stats::rnorm(1.0,2.0,1776);

Parameters

mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.
seed_val – initialize the random engine with a non-negative integral-valued seed.

Returns

a pseudo-random draw from the Normal distribution.

Convenience

template<typename T = double> inline T rnorm()¶

Random sampling function for the standard Normal distribution.

Example:

stats::rnorm();

Returns: a pseudo-random draw from the standard Normal distribution.

Vector/Matrix Output ¶

Random number engines

template<typename mT, typename T1 = double, typename T2 = double> inline mT rnorm(const ullint_t n, const ullint_t k, const T1 mu_par, const T2 sigma_par, rand_engine_t &engine)¶

Random matrix sampling function for the Normal distribution.

Example:

stats::rand_engine_t engine(1776);
// std::vector
stats::rnorm<std::vector<double>>(5,4,1.0,2.0,engine);
// Armadillo matrix
stats::rnorm<arma::mat>(5,4,1.0,2.0,engine);
// Blaze dynamic matrix
stats::rnorm<blaze::DynamicMatrix<double,blaze::columnMajor>>(5,4,1.0,2.0,engine);
// Eigen dynamic matrix
stats::rnorm<Eigen::MatrixXd>(5,4,1.0,2.0,engine);

Note

This function requires template instantiation; acceptable output types include: std::vector, with element type float, double, etc., as well as Armadillo, Blaze, and Eigen dense matrices.

Parameters

n – the number of output rows
k – the number of output columns
mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.
engine – a random engine, passed by reference.

Returns

a matrix of pseudo-random draws from the Normal distribution.

Seed values

template<typename mT, typename T1, typename T2> inline mT rnorm(const ullint_t n, const ullint_t k, const T1 mu_par, const T2 sigma_par, const ullint_t seed_val = std::random_device{}())¶

Random matrix sampling function for the Normal distribution.

Example:

// std::vector
stats::rnorm<std::vector<double>>(5,4,1.0,2.0);
// Armadillo matrix
stats::rnorm<arma::mat>(5,4,1.0,2.0);
// Blaze dynamic matrix
stats::rnorm<blaze::DynamicMatrix<double,blaze::columnMajor>>(5,4,1.0,2.0);
// Eigen dynamic matrix
stats::rnorm<Eigen::MatrixXd>(5,4,1.0,2.0);

Note

This function requires template instantiation; acceptable output types include: std::vector, with element type float, double, etc., as well as Armadillo, Blaze, and Eigen dense matrices.

Parameters

n – the number of output rows
k – the number of output columns
mu_par – the mean parameter, a real-valued input.
sigma_par – the standard deviation parameter, a real-valued input.
seed_val – initialize the random engine with a non-negative integral-valued seed.

Returns

a matrix of pseudo-random draws from the Normal distribution.