binomial.rst

Binomial Distribution

Table of contents

• Density Function
  • Scalar Input
  • Vector/Matrix Input
    • STL Containers
    • Armadillo
    • Blaze
    • Eigen
• Cumulative Distribution Function
  • Scalar Input
  • Vector/Matrix Input
    • STL Containers
    • Armadillo
    • Blaze
    • Eigen
• Quantile Function
  • Scalar Input
  • Vector/Matrix Input
    • STL Containers
    • Armadillo
    • Blaze
    • Eigen
• Random Sampling
  • Scalar Output
  • Vector/Matrix Output

---

Density Function

The density function of the Binomial distribution:

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

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

Scalar Input

.. doxygenfunction:: dbinom(const llint_t, const llint_t, const T, const bool)
   :project: statslib

Vector/Matrix Input

STL Containers

.. doxygenfunction:: dbinom(const std::vector<eT>&, const llint_t, const T1, const bool)
   :project: statslib

Armadillo

.. doxygenfunction:: dbinom(const ArmaMat<eT>&, const llint_t, const T1, const bool)
   :project: statslib

Blaze

.. doxygenfunction:: dbinom(const BlazeMat<eT, To>&, const llint_t, const T1, const bool)
   :project: statslib

Eigen

.. doxygenfunction:: dbinom(const EigenMat<eT, iTr, iTc>&, const llint_t, const T1, const bool)
   :project: statslib

---

Cumulative Distribution Function

The cumulative distribution function of the Binomial distribution:

F(x; n, p) = \sum_{z \leq x} f(z; n, p)

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

Scalar Input

.. doxygenfunction:: pbinom(const llint_t, const llint_t, const T, const bool)
   :project: statslib

Vector/Matrix Input

STL Containers

.. doxygenfunction:: pbinom(const std::vector<eT>&, const llint_t, const T1, const bool)
   :project: statslib

Armadillo

.. doxygenfunction:: pbinom(const ArmaMat<eT>&, const llint_t, const T1, const bool)
   :project: statslib

Blaze

.. doxygenfunction:: pbinom(const BlazeMat<eT, To>&, const llint_t, const T1, const bool)
   :project: statslib

Eigen

.. doxygenfunction:: pbinom(const EigenMat<eT, iTr, iTc>&, const llint_t, const T1, const bool)
   :project: statslib

---

Quantile Function

The quantile function of the Binomial distribution:

q(r; n, p) = \inf \left\{ x : r \leq F(x; n, p) \right\}

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

Scalar Input

.. doxygenfunction:: qbinom(const T1, const llint_t, const T2)
   :project: statslib

Vector/Matrix Input

STL Containers

.. doxygenfunction:: qbinom(const std::vector<eT>&, const llint_t, const T1)
   :project: statslib

Armadillo

.. doxygenfunction:: qbinom(const ArmaMat<eT>&, const llint_t, const T1)
   :project: statslib

Blaze

.. doxygenfunction:: qbinom(const BlazeMat<eT, To>&, const llint_t, const T1)
   :project: statslib

Eigen

.. doxygenfunction:: qbinom(const EigenMat<eT, iTr, iTc>&, const llint_t, const T1)
   :project: statslib

---

Random Sampling

Random sampling for the Binomial distribution is achieved by summing the results of simulating n Bernoulli-distributed random variables.

Scalar Output

1. Random number engines

.. doxygenfunction:: rbinom(const llint_t, const T, rand_engine_t&)
   :project: statslib

2. Seed values

.. doxygenfunction:: rbinom(const llint_t, const T, const ullint_t)
   :project: statslib

Vector/Matrix Output

1. Random number engines

.. doxygenfunction:: rbinom(const ullint_t, const ullint_t, const llint_t, const T1, rand_engine_t&)
   :project: statslib

2. Seed values

.. doxygenfunction:: rbinom(const ullint_t, const ullint_t, const llint_t, const T1, const ullint_t)
   :project: statslib