Distributions

Nautilus.Distributions provides functions for 12 probability distribution families.

Distribution families

Distribution	PDF	CDF	Survival	Inverse CDF	Sample
Normal	normal_pdf	normal_cdf		normal_inv_cdf	normal_sample
LogNormal	lognormal_pdf	lognormal_cdf		lognormal_inv_cdf	lognormal_sample
Uniform	uniform_pdf	uniform_cdf		uniform_inv_cdf	uniform_sample
Exponential	exponential_pdf	exponential_cdf		exponential_inv_cdf	exponential_sample
Gamma	gamma_pdf	gamma_cdf	gamma_sf	gamma_inv_cdf	gamma_sample
Chi-squared	chi_squared_pdf	chi_squared_cdf	chi_squared_sf	chi_squared_inv_cdf	chi_squared_sample
Student-t	student_t_pdf	student_t_cdf			student_t_sample
Poisson	poisson_pmf	poisson_cdf
Binomial	binomial_pmf	binomial_cdf
Beta	beta_pdf	beta_cdf
F	f_pdf	f_cdf
Weibull	weibull_pdf	weibull_cdf		weibull_inv_cdf

Only the gamma family has a survival function. For the other families, get the upper tail from the symmetry of the distribution, if it has one. Do not calculate 1 - cdf, which loses the full tail to 0.5 * ulp(1.0).

• Standard normal: normal_cdf(neg(z), 0, 1) is the exact upper tail at z.
• Student-t: student_t_cdf(neg(t), df) is the exact upper tail at t for all df, because the distribution is symmetric about zero. The identity also applies at df <= 1, where the mean does not exist.
• Non-standard normal: reflect the mean and the point. The upper tail of N(mu, sigma) at x is Phi((mu - x) / sigma), so write normal_cdf(neg(x), neg(mean), std). If you negate only the point, the result is Phi((-x - mu) / sigma), which is a different value. For N(10, 2) at x = 13, that call returns 6.6e-31, but the correct value is 0.0668.

See Right-tail accuracy.

Parameter conventions

• Normal: (x, mean, std), even for the standard normal
• Gamma: (x, shape, scale), SciPy's convention, not (x, shape, rate)
• Exponential: (x, rate), unlike SciPy's scale = 1/rate
• Chi-squared: (x, df) where df is degrees of freedom
• Student-t: (x, df)
• Poisson: (k, lambda) where k is the count
• Binomial: (k, n, p) where n is trials and p is probability
• Beta: (x, a, b) where a, b are shape parameters
• F: (x, d1, d2) where d1 and d2 are the numerator and denominator degrees of freedom
• Weibull: (x, shape, scale) where shape is k and scale is lambda

Discrete counts and trial numbers are f32 values, not integers. Each family page lists its parameter domain and what an invalid parameter returns; several families do not check their parameters at all.

Sampling with explicit keys

The seven _sample functions take a key as their first argument:

def normal_sample[n](k: key, template: tensor[n, f32], mean: f32, std: f32) -> tensor[n, f32]

The template tensor determines the output shape. The actual values in the template are ignored; only its shape is used. See Sampling with explicit keys for key derivation, methods, and the gamma, chi-squared, and Student-t sampling limits.

Imports

import Nautilus.Distributions (normal_pdf, normal_cdf, normal_inv_cdf)
import Nautilus.Distributions (gamma_cdf, student_t_cdf, chi_squared_cdf)
import Nautilus.Distributions (poisson_cdf, binomial_cdf, beta_cdf)

Common usage

normal_cdf and normal_inv_cdf appear in Black-Scholes pricing, value-at-risk computation, and hypothesis testing. gamma_cdf and student_t_cdf underpin chi-squared and t-test p-values through the Nautilus.Testing module.