Bernoulli Distribution¶
Table of contents
Density Function¶
The density function of the Bernoulli distribution:
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
.
Blaze¶
-
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:
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:
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 typefloat
,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 typefloat
,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.