Poisson Distribution¶
Table of contents
Density Function¶
The density function of the Poisson distribution:
Methods for scalar input, as well as for vector/matrix input, are listed below.
Scalar Input¶
-
template<typename T>
constexpr return_t<T> dpois(const llint_t x, const T rate_par, const bool log_form = false) noexcept¶ Density function of the Poisson distribution.
Example:
stats::dpois(8.0,10.0,false);
- Parameters
x – a non-negative integral-valued input.
rate_par – the rate 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> dpois(const std::vector<eT> &x, const T1 rate_par, const bool log_form = false)¶ Density function of the Poisson distribution.
Example:
std::vector<int> x = {2, 3, 4}; stats::dpois(x,4,false);
- Parameters
x – a standard vector.
rate_par – the rate 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 rT = common_return_t<eT, T1>>
inline ArmaMat<rT> dpois(const ArmaMat<eT> &X, const T1 rate_par, const bool log_form = false)¶ Density function of the Poisson distribution.
Example:
arma::mat X = { {2, 1, 4}, {3, 5, 6} }; stats::dpois(X,4,false);
- Parameters
X – a matrix of input values.
rate_par – the rate 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> dpois(const BlazeMat<eT, To> &X, const T1 rate_par, const bool log_form = false)¶ Density function of the Poisson distribution.
Example:
stats::dpois(X,4,false);
- Parameters
X – a matrix of input values.
rate_par – the rate 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> dpois(const EigenMat<eT, iTr, iTc> &X, const T1 rate_par, const bool log_form = false)¶ Density function of the Poisson distribution.
Example:
stats::dpois(X,4,false);
- Parameters
X – a matrix of input values.
rate_par – the rate 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 Poisson distribution:
Methods for scalar input, as well as for vector/matrix input, are listed below.
Scalar Input¶
-
template<typename T>
constexpr return_t<T> ppois(const llint_t x, const T rate_par, const bool log_form = false) noexcept¶ Distribution function of the Poisson distribution.
Example:
stats::ppois(8.0,10.0,false);
- Parameters
x – a non-negative integral-valued input.
rate_par – the rate 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> ppois(const std::vector<eT> &x, const T1 rate_par, const bool log_form = false)¶ Distribution function of the Poisson distribution.
Example:
std::vector<int> x = {2, 3, 4}; stats::ppois(x,2.0,false);
- Parameters
x – a standard vector.
rate_par – the rate 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> ppois(const ArmaMat<eT> &X, const T1 rate_par, const bool log_form = false)¶ Distribution function of the Poisson distribution.
Example:
arma::mat X = { {2, 1, 4}, {3, 5, 6} }; stats::ppois(X,2.0,false);
- Parameters
X – a matrix of input values.
rate_par – the rate 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> ppois(const BlazeMat<eT, To> &X, const T1 rate_par, const bool log_form = false)¶ Distribution function of the Poisson distribution.
Example:
stats::ppois(X,2.0,false);
- Parameters
X – a matrix of input values.
rate_par – the rate 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> ppois(const EigenMat<eT, iTr, iTc> &X, const T1 rate_par, const bool log_form = false)¶ Distribution function of the Poisson distribution.
Example:
stats::ppois(X,2.0,false);
- Parameters
X – a matrix of input values.
rate_par – the rate 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 Poisson 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> qpois(const T1 p, const T2 rate_par) noexcept¶ Quantile function of the Poisson distribution.
Example:
stats::qpois(0.6,10.0);
- Parameters
p – a real-valued input.
rate_par – the rate 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> qpois(const std::vector<eT> &x, const T1 rate_par)¶ Quantile function of the Poisson distribution.
Example:
std::vector<double> x = {0.3, 0.5, 0.8}; stats::qpois(x,4);
- Parameters
x – a standard vector.
rate_par – the rate 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> qpois(const ArmaMat<eT> &X, const T1 rate_par)¶ Quantile function of the Poisson distribution.
Example:
arma::mat X = { {0.2, 0.7, 0.9}, {0.1, 0.8, 0.3} }; stats::qpois(X,4);
- Parameters
X – a matrix of input values.
rate_par – the rate 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> qpois(const BlazeMat<eT, To> &X, const T1 rate_par)¶ Quantile function of the Poisson distribution.
Example:
stats::qpois(X,4);
- Parameters
X – a matrix of input values.
rate_par – the rate 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> qpois(const EigenMat<eT, iTr, iTc> &X, const T1 rate_par)¶ Quantile function of the Poisson distribution.
Example:
stats::qpois(X,4);
- Parameters
X – a matrix of input values.
rate_par – the rate parameter, a real-valued input.
- Returns
a matrix of quantile values corresponding to the elements of
X
.
Random Sampling¶
Scalar Output¶
Random number engines
-
template<typename T>
inline return_t<T> rpois(const T rate_par, rand_engine_t &engine)¶ Random sampling function for the Poisson distribution.
Example:
stats::rand_engine_t engine(1776); stats::rchisq(4,engine);
- Parameters
rate_par – the rate parameter, a real-valued input.
engine – a random engine, passed by reference.
- Returns
a pseudo-random draw from the Poisson distribution.
Seed values
-
template<typename T>
inline return_t<T> rpois(const T rate_par, const ullint_t seed_val = std::random_device{}())¶ Random sampling function for the Poisson distribution.
Example:
stats::rchisq(4,1776);
- Parameters
rate_par – the rate 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 Poisson distribution.
Vector/Matrix Output¶
Random number engines
-
template<typename mT, typename T1>
inline mT rpois(const ullint_t n, const ullint_t k, const T1 rate_par, rand_engine_t &engine)¶ Random matrix sampling function for the Poisson distribution.
Example:
stats::rand_engine_t engine(1776); // std::vector stats::rpois<std::vector<double>>(5,4,4,engine); // Armadillo matrix stats::rpois<arma::mat>(5,4,4,engine); // Blaze dynamic matrix stats::rpois<blaze::DynamicMatrix<double,blaze::columnMajor>>(5,4,4,engine); // Eigen dynamic matrix stats::rpois<Eigen::MatrixXd>(5,4,4,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
rate_par – the rate parameter, a real-valued input.
engine – a random engine, passed by reference.
- Returns
a matrix of pseudo-random draws from the Poisson distribution.
Seed values
-
template<typename mT, typename T1>
inline mT rpois(const ullint_t n, const ullint_t k, const T1 rate_par, const ullint_t seed_val = std::random_device{}())¶ Random matrix sampling function for the Poisson distribution.
Example:
// std::vector stats::rpois<std::vector<double>>(5,4,4); // Armadillo matrix stats::rpois<arma::mat>(5,4,4); // Blaze dynamic matrix stats::rpois<blaze::DynamicMatrix<double,blaze::columnMajor>>(5,4,4); // Eigen dynamic matrix stats::rpois<Eigen::MatrixXd>(5,4,4);
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
rate_par – the rate 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 Poisson distribution.