# Normal Inverse Gaussian Distribution¶

The probability density function is given by:

\begin{eqnarray*} f(x; a, b) = \frac{a \exp\left(\sqrt{a^2 - b^2} + b x \right)}{\pi \sqrt{1 + x^2}} \, K_1\left(a * sqrt{1 + x^2}\right), \end{eqnarray*}

where $$x$$ is a real number, the parameter $$a$$ is the tail heaviness and $$b$$ is the asymmetry parameter satisfying $$a > 0$$ and $$|b| \leq a$$. $$K_1$$ is the modified Bessel function of second kind (scipy.special.k1).

A normal inverse Gaussian random variable with parameters $$a$$ and $$b$$ can be expressed as $$X = b V + \sqrt(V) X$$ where $$X$$ is norm(0,1) and $$V$$ is invgauss(mu=1/sqrt(a**2 - b**2)). Hence, the normal inverse Gaussian distribution is a special case of normal variance-mean mixtures.

Implementation: scipy.stats.norminvgauss

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