If shape, skewness, location and scale are not specified they assume the default values of 1, 0, 0 and 1, respectively.
The Normal-inverse Gaussian distribution with parameters shape = \(\alpha\),
skewness = \(\beta\), location = \(\mu\) and scale = \(\delta\) has density:
$$ \frac{\alpha\delta K_1(\alpha\sqrt{\delta^2+(x-\mu)^2})}{\pi\sqrt{\delta^2+(x-\mu)^2}}e^{\delta\gamma+\beta(x-\mu)} $$
where \(\gamma = \sqrt(\alpha^2 - \beta^2)\) and
\(K_1\) denotes a modified Bessel function of the second kind.
The mean and variance of NIG are defined respectively \(\mu + \beta \delta / \gamma\) and
\(\delta \alpha^2 / \gamma^3\).