# Random numbers and distributions


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``` python
import rustymath as rm
```

## A persistent generator

`default_rng(seed)` creates one generator whose state advances across
calls. A size tuple gives the output shape. Reuse the generator for
different distributions:

``` python
rng = rm.default_rng(42)
draws = rng.normal(size=(3, 2))
draws, draws.mean(axis=0), rng.sample(rm.distribution.beta(2, 3), 5)
```

    (Array([[-1.0920367, -0.30040901], [-0.78924237, 0.52985647], [-2.2347094, 1.3224215]], dtype=float64),
     Array([-1.3719962, 0.51728964], dtype=float64),
     Array([0.16794595, 0.31556707, 0.25183041, 0.32702751, 0.60316242], dtype=float64))

`integers` draws from a half-open interval. `choice` and `permutation`
operate on leading-axis cells; an integer argument denotes a range. Here
the two selected items are whole rows:

``` python
rng.integers(0, 10, size=4), rng.choice(draws, size=2, replace=False)
```

    (Array([5, 3, 0, 4], dtype=int64),
     Array([[-1.0920367, -0.30040901], [-2.2347094, 1.3224215]], dtype=float64))

The generator uses rustymath’s shared RNG. It does not reproduce NumPy’s
random sequences.

`multivariate_normal(mean, cov, size)` correlates ordinary normal draws
through a Cholesky factor and a matrix product. It takes one real mean
vector and real positive-definite covariance matrix, both without axis
keys or names, returning float64 samples with a final component axis. A
covariance that cannot be factored follows `cholesky`’s NaN-result
policy.

``` python
rng.multivariate_normal([1., 2.], [[4., 1.], [1., 3.]], size=3)
```

    Array([[-0.90930908, 2.008366], [4.0249491, 6.4071364], [-1.7609395, 1.433404]], dtype=float64)

## Probability functions

`rm.distribution` holds sixteen distributions. Each constructor takes
the distribution’s parameters, or uses standard ones where the
distribution has them. The result has `pdf`, `cdf`, `quantile` and
`sample`. The same `seed` gives the same draws:

``` python
n = rm.distribution.normal()
n.cdf([-1.96, 0, 1.96]), n.quantile(0.975), n.sample(3, seed=0)
```

    (Array([0.024997895, 0.5, 0.9750021], dtype=float64),
     1.9599639845400538,
     Array([0.11623959, -0.78285485, -2.3346353], dtype=float64))