Bernoulli Distribution¶

Table of contents

Density Function ¶

The density function of the Bernoulli distribution:

\[f(x; p) = p^x (1-p)^{1-x} \times \mathbf{1}[x \in \{0,1\}]\]

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

Scalar Input ¶

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

Density function of the Bernoulli distribution.

Example:

stats::dbern(1,0.6,false); 

Parameters

x – an integral-valued input, equal to 0 or 1.
prob_par – the probability parameter, a real-valued input.
log_form – return the log-density or the true form.

Returns

the density function evaluated at x.

Vector/Matrix Input ¶

STL Containers ¶

template<typename eT, typename T1, typename rT = common_return_t<eT, T1>> inline std::vector<rT> dbern(const std::vector<eT> &x, const T1 prob_par, const bool log_form = false)¶

Density function of the Bernoulli distribution.

Example:

std::vector<int> x = {0, 1, 0};
stats::dbern(x,0.5,false);

Parameters

x – a standard vector.
prob_par – the probability 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 mT, typename tT, typename T1> inline mT dbern(const ArmaGen<mT, tT> &X, const T1 prob_par, const bool log_form = false)¶

Density function of the Bernoulli distribution.

Example:

arma::mat X = { {1, 0, 1},
                {0, 1, 0} };
stats::dbern(X,0.5,false);

Parameters

X – a matrix of input values.
prob_par – the probability 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.

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

Density function of the Bernoulli distribution.

Example:

stats::dbern(X,0.5,false);

Parameters

X – a matrix of input values.
prob_par – the probability 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 rT = common_return_t<eT, T1>, int iTr = Eigen::Dynamic, int iTc = Eigen::Dynamic> inline EigenMat<rT, iTr, iTc> dbern(const EigenMat<eT, iTr, iTc> &X, const T1 prob_par, const bool log_form = false)¶

Density function of the Bernoulli distribution.

Example:

stats::dbern(X,0.5,false);

Parameters

X – a matrix of input values.
prob_par – the probability 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 Bernoulli distribution:

\[F(x; p) = \sum_{z \leq x} f(z; p) = \begin{cases} 0 & \text{ if } x < 0 \\ 1-p & \text{ if } 0 \leq x < 1 \\ 1 & \text{ if } x \geq 1 \end{cases}\]

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

Scalar Input ¶

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

Distribution function of the Bernoulli distribution.

Example:

stats::pbern(1,0.6,false); 

Parameters

x – a value equal to 0 or 1.
prob_par – the probability parameter, 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 rT = common_return_t<eT, T1>> inline std::vector<rT> pbern(const std::vector<eT> &x, const T1 prob_par, const bool log_form = false)¶

Density function of the Bernoulli distribution.

Example:

std::vector<int> x = {0, 1, 0};
stats::pbern(x,0.5,false);

Parameters

x – a standard vector.
prob_par – the probability 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 rT = common_return_t<eT, T1>> inline ArmaMat<rT> pbern(const ArmaMat<eT> &X, const T1 prob_par, const bool log_form = false)¶

Density function of the Bernoulli distribution.

Example:

arma::mat X = { {1, 0, 1},
                {0, 1, 0} };
stats::pbern(X,0.5,false);

Parameters

X – a matrix of input values.
prob_par – the probability 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 rT = common_return_t<eT, T1>, bool To = blaze::columnMajor> inline BlazeMat<rT, To> pbern(const BlazeMat<eT, To> &X, const T1 prob_par, const bool log_form = false)¶

Density function of the Bernoulli distribution.

Example:

stats::pbern(X,0.5,false);

Parameters

X – a matrix of input values.
prob_par – the probability 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 rT = common_return_t<eT, T1>, int iTr = Eigen::Dynamic, int iTc = Eigen::Dynamic> inline EigenMat<rT, iTr, iTc> pbern(const EigenMat<eT, iTr, iTc> &X, const T1 prob_par, const bool log_form = false)¶

Density function of the Bernoulli distribution.

Example:

stats::pbern(X,0.5,false);

Parameters

X – a matrix of input values.
prob_par – the probability 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 Bernoulli distribution:

\[q(r; p) = \begin{cases} 0 & \text{ if } r \leq 1 - p \\ 1 & \text{ else } \end{cases}\]

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

Scalar Input ¶

template<typename T1, typename T2> constexpr common_return_t<T1, T2> qbern(const T1 p, const T2 prob_par) noexcept¶

Quantile function of the Bernoulli distribution.

Example:

stats::qbern(0.5,0.4); 

Parameters

p – a real-valued input.
prob_par – the probability parameter, a real-valued input.

Returns

the quantile function evaluated at p.

Vector/Matrix Input ¶

STL Containers ¶

template<typename eT, typename T1, typename rT = common_return_t<eT, T1>> inline std::vector<rT> qbern(const std::vector<eT> &x, const T1 prob_par)¶

Quantile function of the Bernoulli distribution.

Example:

std::vector<int> x = {0.4, 0.5, 0.9};
stats::qbern(x,0.5);

Parameters

x – a standard vector.
prob_par – the probability parameter, a real-valued input.

Returns

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

Armadillo ¶

template<typename eT, typename T1, typename rT = common_return_t<eT, T1>> inline ArmaMat<rT> qbern(const ArmaMat<eT> &X, const T1 prob_par)¶

Quantile function of the Bernoulli distribution.

Example:

arma::mat X = { {0.4, 0.5, 0.9},
                {0.3, 0.6, 0.7} };
stats::qbern(X,0.5);

Parameters

X – a matrix of input values.
prob_par – the probability parameter, a real-valued input.

Returns

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

Blaze ¶

template<typename eT, typename T1, typename rT = common_return_t<eT, T1>, bool To = blaze::columnMajor> inline BlazeMat<rT, To> qbern(const BlazeMat<eT, To> &X, const T1 prob_par)¶

Quantile function of the Bernoulli distribution.

Example:

stats::qbern(X,0.5);

Parameters

X – a matrix of input values.
prob_par – the probability parameter, a real-valued input.

Returns

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

Eigen ¶

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

Quantile function of the Bernoulli distribution.

Example:

stats::qbern(X,0.5);

Parameters

X – a matrix of input values.
prob_par – the probability parameter, a real-valued input.

Returns

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

Random Sampling ¶

Random sampling for the Bernoulli distribution is achieved via the inverse probability integral transform.

Scalar Output ¶

Random number engines

template<typename T> inline return_t<T> rbern(const T prob_par, rand_engine_t &engine)¶

Random sampling function for the Bernoulli distribution.

Example:

stats::rand_engine_t engine(1776);
stats::rbern(0.7,engine);

Parameters

prob_par – the probability parameter, a real-valued input.
engine – a random engine, passed by reference.

Returns

a pseudo-random draw from the Bernoulli distribution.

Seed values

template<typename T> inline return_t<T> rbern(const T prob_par, const ullint_t seed_val = std::random_device{}())¶

Random sampling function for the Bernoulli distribution.

Example:

stats::rbern(0.7,1776);

Parameters

prob_par – the probability 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 Bernoulli distribution.

Vector/Matrix Output ¶

Random number engines

template<typename mT, typename T1> inline mT rbern(const ullint_t n, const ullint_t k, const T1 prob_par, rand_engine_t &engine)¶

Random matrix sampling function for the Bernoulli distribution.

Example:

stats::rand_engine_t engine(1776);
// std::vector
stats::rbern<std::vector<double>>(5,4,0.7,engine);
// Armadillo matrix
stats::rbern<arma::mat>(5,4,0.7,engine);
// Blaze dynamic matrix
stats::rbern<blaze::DynamicMatrix<double,blaze::columnMajor>>(5,4,0.7,engine);
// Eigen dynamic matrix
stats::rbern<Eigen::MatrixXd>(5,4,0.7,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
prob_par – the probability parameter, a real-valued input.
engine – a random engine, passed by reference.

Returns

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

Seed values

template<typename mT, typename T1> inline mT rbern(const ullint_t n, const ullint_t k, const T1 prob_par, const ullint_t seed_val = std::random_device{}())¶

Random matrix sampling function for the Bernoulli distribution.

Example:

// std::vector
stats::rbern<std::vector<double>>(5,4,0.7);
// Armadillo matrix
stats::rbern<arma::mat>(5,4,0.7);
// Blaze dynamic matrix
stats::rbern<blaze::DynamicMatrix<double,blaze::columnMajor>>(5,4,0.7);
// Eigen dynamic matrix
stats::rbern<Eigen::MatrixXd>(5,4,0.7);

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
prob_par – the probability 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 Bernoulli distribution.