Overview

Nautilus.Distributions provides 12 probability distribution families. It replaces scipy.stats for Chelis programs.

Distribution families

Distribution	PDF	CDF	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_inv_cdf`
Chi-squared	`chi_squared_pdf`	`chi_squared_cdf`	`chi_squared_inv_cdf`
Student-t	`student_t_pdf`	`student_t_cdf`
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`

Parameter conventions

Functions follow scipy's parameter naming:

Normal: (x, mean, std) not (x, mu, sigma)
Gamma: (x, shape, scale) not (x, shape, rate)
Exponential: (x, 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, df1, df2)
Weibull: (x, shape, scale)

Sampling and the Random effect

The four _sample functions carry the ! { Random } effect:

def normal_sample[n](template: tensor[n, f32], mean: f32, std: f32) -> tensor[n, f32] ! { Random }

The template tensor determines the output shape. The actual values in the template are ignored; only its shape is used. Sampling uses Box-Muller (Normal), inverse CDF (Exponential, LogNormal), or direct scaling (Uniform).

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

The most frequently used functions are normal_cdf and normal_inv_cdf, which 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.