...one of the most highly
regarded and expertly designed C++ library projects in the
world.
— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards
#include <boost/math/distributions/bernoulli.hpp>
namespace boost{ namespace math{ template <class RealType = double, class Policy = policies::policy<> > class bernoulli_distribution; typedef bernoulli_distribution<> bernoulli; template <class RealType, class Policy> class bernoulli_distribution { public: typedef RealType value_type; typedef Policy policy_type; bernoulli_distribution(RealType p); // Constructor. // Accessor function. 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.
probability density function pdf
f(0) = 1  p, f(1) = p
Cumulative distribution function
D(k) = if (k == 0) 1  p else 1
The following graph illustrates how the probability density function pdf varies with the outcome of the single trial:
and the Cumulative distribution function
bernoulli_distribution(RealType p);
Constructs a bernoulli distribution with success_fraction p.
RealType success_fraction() const
Returns the success_fraction parameter of this distribution.
All the usual nonmember accessor functions that are generic to all distributions are supported: Cumulative Distribution Function, Probability Density Function, Quantile, Hazard Function, Cumulative Hazard Function, mean, median, mode, variance, standard deviation, skewness, kurtosis, kurtosis_excess, range and support.
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.
The Bernoulli distribution is implemented with simple arithmetic operators and so should have errors within an epsilon or two.
In the following table p is the probability of success and q = 1p. 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:

Function 
Implementation Notes 

Supported range 
{0, 1} 

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 <= (1p) 0 else 1 
quantile from the complement 
if x <= (1p) 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 