[section:bernoulli_dist Bernoulli Distribution]

``#include <boost/math/distributions/bernoulli.hpp>``

   namespace boost{ namespace math{
    template <class RealType = double,
              class ``__Policy``   = ``__policy_class`` >
    class bernoulli_distribution;

    typedef bernoulli_distribution<> bernoulli;

    template <class RealType, class ``__Policy``>
    class bernoulli_distribution
    {
    public:
       typedef RealType  value_type;
       typedef Policy    policy_type;

       BOOST_MATH_GPU_ENABLED bernoulli_distribution(RealType p); // Constructor.
       // Accessor function.
       BOOST_MATH_GPU_ENABLED RealType success_fraction() const
       // Probability of success (as a fraction).
    };
   }} // namespaces

The Bernoulli distribution is a discrete distribution of the outcome
of a single trial with only two results, 0 (failure) or 1 (success),
with a probability of success p.

The Bernoulli distribution is the simplest building block
on which other discrete distributions of
sequences of independent Bernoulli trials can be based.

The Bernoulli is the binomial distribution (k = 1, p) with only one trial.

[@http://en.wikipedia.org/wiki/Probability_density_function probability density function pdf]
[expression f(0) = 1 - p, f(1) = p]

[@http://en.wikipedia.org/wiki/Cumulative_Distribution_Function Cumulative distribution function]
[expression D(k) = if (k == 0) 1 - p else 1]

The following graph illustrates how the
[@http://en.wikipedia.org/wiki/Probability_density_function probability density function pdf]
varies with the outcome of the single trial:

[graph bernoulli_pdf]

and the [@http://en.wikipedia.org/wiki/Cumulative_Distribution_Function Cumulative distribution function]

[graph bernoulli_cdf]

[h4 Member Functions]

   BOOST_MATH_GPU_ENABLED bernoulli_distribution(RealType p);

Constructs a [@http://en.wikipedia.org/wiki/bernoulli_distribution
bernoulli distribution] with success_fraction /p/.

   BOOST_MATH_GPU_ENABLED RealType success_fraction() const

Returns the /success_fraction/ parameter of this distribution.

[h4 Non-member Accessors]

All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
that are generic to all distributions are supported: __usual_accessors.
For this distribution all non-member accessor functions are marked with `BOOST_MATH_GPU_ENABLED` and can
be run on both host and device.

The domain of the random variable is 0 and 1,
and the useful supported range is only 0 or 1.

Outside this range, functions are undefined, or may throw domain_error exception
and make an error message available.

[h4 Accuracy]

The Bernoulli distribution is implemented with simple arithmetic operators
and so should have errors within an epsilon or two.

[h4 Implementation]

In the following table /p/ is the probability of success and /q = 1-p/.
/k/ is the random variate, either 0 or 1.

[note The Bernoulli distribution is implemented here as a /strict discrete/ distribution.
If a generalised version, allowing k to be any real, is required then
the binomial distribution with a single trial should be used, for example:

`binomial_distribution(1, 0.25)`
]

[table
[[Function][Implementation Notes]]
[[Supported range][{0, 1}]]
[[pdf][Using the relation: pdf = 1 - p for k = 0, else p ]]
[[cdf][Using the relation: cdf = 1 - p for k = 0, else 1]]
[[cdf complement][q = 1 - p]]
[[quantile][if x <= (1-p) 0 else 1]]
[[quantile from the complement][if x <= (1-p) 1 else 0]]
[[mean][p]]
[[variance][p * (1 - p)]]
[[mode][if (p < 0.5) 0 else 1]]
[[skewness][(1 - 2 * p) / sqrt(p * q)]]
[[kurtosis][6 * p * p - 6 * p +1/ p * q]]
[[kurtosis excess][kurtosis -3]]
]

[h4 References]
* [@http://en.wikipedia.org/wiki/Bernoulli_distribution Wikipedia Bernoulli distribution]
* [@http://mathworld.wolfram.com/BernoulliDistribution.html Weisstein, Eric W. "Bernoulli Distribution." From MathWorld--A Wolfram Web Resource.]

[endsect] [/section:bernoulli_dist bernoulli]

[/
  Copyright 2006 John Maddock and Paul A. Bristow.
  Distributed under the Boost Software License, Version 1.0.
  (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt).
]

