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#include <boost/math/distributions/exponential.hpp>
template <class RealType = double, class Policy = policies::policy<> > class exponential_distribution; typedef exponential_distribution<> exponential; template <class RealType, class Policy> class exponential_distribution { public: typedef RealType value_type; typedef Policy policy_type; BOOST_MATH_GPU_ENABLED exponential_distribution(RealType lambda = 1); BOOST_MATH_GPU_ENABLED RealType lambda()const; };
The exponential distribution is a continuous probability distribution with PDF:
It is often used to model the time between independent events that happen at a constant average rate.
The following graph shows how the distribution changes for different values of the rate parameter lambda:
BOOST_MATH_GPU_ENABLED exponential_distribution(RealType lambda = 1);
Constructs an Exponential distribution with parameter lambda. Lambda is defined as the reciprocal of the scale parameter.
Requires lambda > 0, otherwise calls domain_error.
BOOST_MATH_GPU_ENABLED RealType lambda()const;
Accessor function returns the lambda parameter of the distribution.
All the usual non-member accessor
functions that are generic to all distributions are supported:
Cumulative Distribution Function,
Probability Density Function,
Quantile, Hazard Function, Cumulative Hazard Function,
__logcdf, __logpdf, mean,
median, mode,
variance, standard deviation, skewness, kurtosis,
kurtosis_excess,
range and support. 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, +∞].
In this distribution the implementation of both logcdf,
and logpdf are specialized
to improve numerical accuracy.
The exponential distribution is implemented in terms of the standard library
functions exp, log, log1p
and expm1 and as such should
have very low error rates.
In the following table λ is the parameter lambda of the distribution, x is the random variate, p is the probability and q = 1-p.
|
Function |
Implementation Notes |
|---|---|
|
|
Using the relation: pdf = λ * exp(-λ * x) |
|
logpdf |
log(pdf) = -expm1(-x * λ) |
|
cdf |
Using the relation: p = 1 - exp(-x * λ) = -expm1(-x * λ) |
|
logcdf |
log(cdf) = log1p(-exp(-x * λ)) |
|
cdf complement |
Using the relation: q = exp(-x * λ) |
|
quantile |
Using the relation: x = -log(1-p) / λ = -log1p(-p) / λ |
|
quantile from the complement |
Using the relation: x = -log(q) / λ |
|
mean |
1/λ |
|
standard deviation |
1/λ |
|
mode |
0 |
|
skewness |
2 |
|
kurtosis |
9 |
|
kurtosis excess |
6 |
(See also the reference documentation for the related Extreme Distributions.)