...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/inverse_gamma.hpp>
namespace boost{ namespace math{ template <class RealType = double, class Policy = policies::policy<> > class inverse_gamma_distribution { public: typedef RealType value_type; typedef Policy policy_type; inverse_gamma_distribution(RealType shape, RealType scale = 1) RealType shape()const; RealType scale()const; }; }} // namespaces
The inverse_gamma distribution is a continuous probability distribution of the reciprocal of a variable distributed according to the gamma distribution.
The inverse_gamma distribution is used in Bayesian statistics.
See inverse gamma distribution.
R inverse gamma distribution functions.
Wolfram inverse gamma distribution.
See also Gamma Distribution.
Note  

In spite of potential confusion with the inverse gamma function, this distribution does provide the typedef: typedef inverse_gamma_distribution<double> gamma;
If you want a boost::math::inverse_gamma_distribution<>
or you can write 
For shape parameter α and scale parameter β, it is defined by the probability density function (PDF):
f(x;α, β) = β^{α} * (1/x) ^{α+1} exp(β/x) / Γ(α)
and cumulative density function (CDF)
F(x;α, β) = Γ(α, β/x) / Γ(α)
The following graphs illustrate how the PDF and CDF of the inverse gamma distribution varies as the parameters vary:
inverse_gamma_distribution(RealType shape = 1, RealType scale = 1);
Constructs an inverse gamma distribution with shape α and scale β.
Requires that the shape and scale parameters are greater than zero, otherwise calls domain_error.
RealType shape()const;
Returns the α shape parameter of this inverse gamma distribution.
RealType scale()const;
Returns the β scale parameter of this inverse gamma 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 variate is [0,+∞].
Note  

Unlike some definitions, this implementation supports a random variate equal to zero as a special case, returning zero for pdf and cdf. 
The inverse gamma distribution is implemented in terms of the incomplete gamma functions gamma_p and gamma_q and their inverses gamma_p_inv and gamma_q_inv: refer to the accuracy data for those functions for more information. But in general, inverse_gamma results are accurate to a few epsilon, >14 decimal digits accuracy for 64bit double.
In the following table α is the shape parameter of the distribution, α is its scale parameter, x is the random variate, p is the probability and q = 1p.
Function 
Implementation Notes 


Using the relation: pdf = gamma_p_derivative(α, β/ x, β) / x * x 
cdf 
Using the relation: p = gamma_q(α, β / x) 
cdf complement 
Using the relation: q = gamma_p(α, β / x) 
quantile 
Using the relation: x = β/ gamma_q_inv(α, p) 
quantile from the complement 
Using the relation: x = α/ gamma_p_inv(α, q) 
mode 
β / (α + 1) 
median 
no analytic equation is known, but is evaluated as quantile(0.5) 
mean 
β / (α  1) for α > 1, else a domain_error 
variance 
(β * β) / ((α  1) * (α  1) * (α  2)) for α >2, else a domain_error 
skewness 
4 * sqrt (α 2) / (α 3) for α >3, else a domain_error 
kurtosis_excess 
(30 * α  66) / ((α3)*(α  4)) for α >4, else a domain_error 