inline RealType pdf(const inverse_gaussian_distribution<RealType, Policy>& dist, const RealType& x)
{ // Probability Density Function
BOOST_MATH_STD_USING // for ADL of std functions
RealType scale = dist.scale();
RealType mean = dist.mean();
RealType result = 0;
static const char* function = "boost::math::pdf(const inverse_gaussian_distribution<%1%>&, %1%)";
if(false == detail::check_scale(function, scale, &result, Policy()))
{
return result;
}
if(false == detail::check_location(function, mean, &result, Policy()))
{
return result;
}
if(false == detail::check_positive_x(function, x, &result, Policy()))
{
return result;
}
if (x == 0)
{
return 0; // Convenient, even if not defined mathematically.
}
result =
sqrt(scale / (constants::two_pi<RealType>() * x * x * x))
* exp(-scale * (x - mean) * (x - mean) / (2 * x * mean * mean));
return result;
} // pdf
inline RealType cdf(const inverse_gaussian_distribution<RealType, Policy>& dist, const RealType& x)
{ // Cumulative Density Function.
BOOST_MATH_STD_USING // for ADL of std functions.
RealType scale = dist.scale();
RealType mean = dist.mean();
static const char* function = "boost::math::cdf(const inverse_gaussian_distribution<%1%>&, %1%)";
RealType result = 0;
if(false == detail::check_scale(function, scale, &result, Policy()))
{
return result;
}
if(false == detail::check_location(function, mean, &result, Policy()))
{
return result;
}
if (false == detail::check_x_gt0(function, mean, &result, Policy()))
{
return result;
}
if(false == detail::check_positive_x(function, x, &result, Policy()))
{
return result;
}
if (x == 0)
{
return 0; // Convenient, even if not defined mathematically.
}
// Problem with this formula for large scale > 1000 or small x,
//result = 0.5 * (erf(sqrt(scale / x) * ((x / mean) - 1) / constants::root_two<RealType>(), Policy()) + 1)
// + exp(2 * scale / mean) / 2
// * (1 - erf(sqrt(scale / x) * (x / mean + 1) / constants::root_two<RealType>(), Policy()));
// so use normal distribution version:
// Wikipedia CDF equation http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution.
normal_distribution<RealType> n01;
RealType n0 = sqrt(scale / x);
n0 *= ((x / mean) -1);
RealType n1 = cdf(n01, n0);
RealType expfactor = exp(2 * scale / mean);
RealType n3 = - sqrt(scale / x);
n3 *= (x / mean) + 1;
RealType n4 = cdf(n01, n3);
result = n1 + expfactor * n4;
return result;
} // cdf
