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# InverseGaussian

The **Inverse Gaussian** (Wald) distribution on $(0, \infty)$ with mean
$\mu > 0$ and shape $\lambda > 0$:

$$
f(x \mid \mu, \lambda) = \sqrt{\frac{\lambda}{2\pi}}\, x^{-3/2}
\exp\!\left(-\frac{\lambda(x - \mu)^2}{2\mu^2 x}\right), \qquad x > 0.
$$

It is exactly $\mathrm{GIG}(p = -\tfrac12,\; a = \lambda/\mu^2,\; b = \lambda)$
and the subordinator behind the {doc}`NormalInverseGaussian
<normal_inverse_gaussian>` mixture — the workhorse heavy-tailed model for
financial returns.

## Density gallery

The shape $\lambda$ controls concentration: as $\lambda \to \infty$ the density
concentrates around $\mu$; small $\lambda$ produces a heavy, right-skewed tail.

```{code-cell} python
:tags: [hide-input]
import jax
jax.config.update("jax_enable_x64", True)
import jax.numpy as jnp
import numpy as np
import matplotlib.pyplot as plt

from normix import InverseGaussian
from normix.utils.plotting import set_theme

set_theme()
np.set_printoptions(precision=5, suppress=True)

x = jnp.linspace(1e-3, 4.0, 400)
fig, ax = plt.subplots()
for lam in [0.5, 1.0, 3.0, 8.0]:
    g = InverseGaussian(mu=jnp.array(1.0), lam=jnp.array(lam))
    ax.plot(np.asarray(x), np.asarray(jax.vmap(g.pdf)(x)), label=rf"$\lambda$ = {lam:g}")
ax.set_xlabel("x"); ax.set_ylabel("density")
ax.set_title(r"InverseGaussian densities ($\mu$ = 1)")
ax.legend()
plt.show()
```

## Parametrizations

Stored attributes: `mu`, `lam`. Exponential family with sufficient statistic
$t(x) = [x,\; 1/x]$:

| Parametrization | Value |
|---|---|
| Classical | mean $\mu > 0$, shape $\lambda > 0$ |
| Natural $\theta$ | $[\,-\lambda/(2\mu^2),\; -\lambda/2\,]$ |
| Expectation $\eta = \nabla\psi$ | $[\,\mu,\; 1/\mu + 1/\lambda\,] = (\mathbb{E}[X],\; \mathbb{E}[1/X])$ |

The $\eta \mapsto \theta$ inversion is closed-form: $\mu = \eta_1$,
$\lambda = 1/(\eta_2 - 1/\eta_1)$.

## Quick usage

```{code-cell} python
ig = InverseGaussian(mu=jnp.array(1.0), lam=jnp.array(2.0))

print("mean/var/std:", float(ig.mean()), float(ig.var()), float(ig.std()))
print("cdf(1.0):    ", float(ig.cdf(jnp.array(1.0))))
print("5% quantile: ", float(ig.ppf(jnp.array(0.05))))

samples = ig.rvs(10_000, seed=0)           # Michael-Schucany-Haas sampler
fitted = InverseGaussian.fit_mle(samples)  # closed-form MLE
print("fit (mu,lam):", float(fitted.mu), float(fitted.lam))
```

## See also

- API: {py:class}`~normix.distributions.inverse_gaussian.InverseGaussian`
- Theory: {doc}`../theory/gig` — InverseGaussian as the $p = -\tfrac12$ GIG
- Tutorial: {doc}`../tutorials/distributions/01_univariate_positive`