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119 lines
3.8 KiB
Plaintext
119 lines
3.8 KiB
Plaintext
[section:bernoulli_dist Bernoulli Distribution]
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``#include <boost/math/distributions/bernoulli.hpp>``
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namespace boost{ namespace math{
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template <class RealType = double,
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class ``__Policy`` = ``__policy_class`` >
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class bernoulli_distribution;
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typedef bernoulli_distribution<> bernoulli;
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template <class RealType, class ``__Policy``>
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class bernoulli_distribution
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{
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public:
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typedef RealType value_type;
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typedef Policy policy_type;
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bernoulli_distribution(RealType p); // Constructor.
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// Accessor function.
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RealType success_fraction() const
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// Probability of success (as a fraction).
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};
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}} // namespaces
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The Bernoulli distribution is a discrete distribution of the outcome
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of a single trial with only two results, 0 (failure) or 1 (success),
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with a probability of success p.
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The Bernoulli distribution is the simplest building block
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on which other discrete distributions of
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sequences of independent Bernoulli trials can be based.
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The Bernoulli is the binomial distribution (k = 1, p) with only one trial.
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[@http://en.wikipedia.org/wiki/Probability_density_function probability density function pdf]
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f(0) = 1 - p, f(1) = p.
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[@http://en.wikipedia.org/wiki/Cumulative_Distribution_Function Cumulative distribution function]
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D(k) = if (k == 0) 1 - p else 1.
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The following graph illustrates how the
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[@http://en.wikipedia.org/wiki/Probability_density_function probability density function pdf]
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varies with the outcome of the single trial:
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[graph bernoulli_pdf]
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and the [@http://en.wikipedia.org/wiki/Cumulative_Distribution_Function Cumulative distribution function]
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[graph bernoulli_cdf]
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[h4 Member Functions]
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bernoulli_distribution(RealType p);
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Constructs a [@http://en.wikipedia.org/wiki/bernoulli_distribution
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bernoulli distribution] with success_fraction /p/.
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RealType success_fraction() const
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Returns the /success_fraction/ parameter of this distribution.
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[h4 Non-member Accessors]
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All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
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that are generic to all distributions are supported: __usual_accessors.
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The domain of the random variable is 0 and 1,
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and the useful supported range is only 0 or 1.
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Outside this range, functions are undefined, or may throw domain_error exception
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and make an error message available.
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[h4 Accuracy]
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The Bernoulli distribution is implemented with simple arithmetic operators
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and so should have errors within an epsilon or two.
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[h4 Implementation]
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In the following table /p/ is the probability of success and /q = 1-p/.
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/k/ is the random variate, either 0 or 1.
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[note The Bernoulli distribution is implemented here as a /strict discrete/ distribution.
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If a generalised version, allowing k to be any real, is required then
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the binomial distribution with a single trial should be used, for example:
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`binomial_distribution(1, 0.25)`
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]
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[table
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[[Function][Implementation Notes]]
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[[Supported range][{0, 1}]]
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[[pdf][Using the relation: pdf = 1 - p for k = 0, else p ]]
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[[cdf][Using the relation: cdf = 1 - p for k = 0, else 1]]
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[[cdf complement][q = 1 - p]]
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[[quantile][if x <= (1-p) 0 else 1]]
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[[quantile from the complement][if x <= (1-p) 1 else 0]]
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[[mean][p]]
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[[variance][p * (1 - p)]]
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[[mode][if (p < 0.5) 0 else 1]]
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[[skewness][(1 - 2 * p) / sqrt(p * q)]]
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[[kurtosis][6 * p * p - 6 * p +1/ p * q]]
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[[kurtosis excess][kurtosis -3]]
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]
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[h4 References]
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* [@http://en.wikipedia.org/wiki/Bernoulli_distribution Wikpedia Bernoulli distribution]
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* [@http://mathworld.wolfram.com/BernoulliDistribution.html Weisstein, Eric W. "Bernoulli Distribution." From MathWorld--A Wolfram Web Resource.]
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[endsect] [/section:bernoulli_dist bernoulli]
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[/
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Copyright 2006 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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