Other continuous (Exponential, Weibull, Beta, F)

Six additional continuous distribution families: Exponential, LogNormal, Uniform, Weibull, Beta, and F. Each provides PDF and CDF; some also provide inverse CDF and sampling.

Exponential

Parameterized by rate (not scale). PDF: rate * exp(-rate * x) for x >= 0.

• exponential_pdf(x: f32, rate: f32) -> f32
• exponential_cdf(x: f32, rate: f32) -> f32: 1 - exp(-rate * x)
• exponential_inv_cdf(q: f32, rate: f32) -> f32: -ln(1 - q) / rate
• exponential_sample[n](template, rate) -> tensor ! { Random }

import Nautilus.Distributions (exponential_cdf)

cdf = exponential_cdf(cast(1.0, f32), cast(2.0, f32))  -- approximately 0.8647

LogNormal

Parameterized by mu and sigma of the underlying normal. Delegates to normal_cdf/normal_inv_cdf after taking log(x).

• lognormal_pdf(x: f32, mu: f32, sigma: f32) -> f32: 0 for x <= 0
• lognormal_cdf(x: f32, mu: f32, sigma: f32) -> f32: normal_cdf(ln(x), mu, sigma)
• lognormal_inv_cdf(q: f32, mu: f32, sigma: f32) -> f32: exp(normal_inv_cdf(q))
• lognormal_sample[n](template, mu, sigma) -> tensor ! { Random }

import Nautilus.Distributions (lognormal_cdf)

cdf = lognormal_cdf(cast(1.0, f32), cast(0.0, f32), cast(1.0, f32))  -- approximately 0.5

Uniform

Parameterized by lo and hi endpoints. 1/(hi - lo) inside, 0 outside.

• uniform_pdf(x: f32, lo: f32, hi: f32) -> f32
• uniform_cdf(x: f32, lo: f32, hi: f32) -> f32
• uniform_inv_cdf(q: f32, lo: f32, hi: f32) -> f32: lo + q*(hi-lo)
• uniform_sample[n](template, lo, hi) -> tensor ! { Random }

import Nautilus.Distributions (uniform_cdf)

cdf = uniform_cdf(cast(0.25, f32), cast(0.0, f32), cast(1.0, f32)) -- 0.25

Weibull

Parameterized by shape (k) and scale (lambda). CDF is closed-form.

• weibull_pdf(x: f32, shape: f32, scale: f32) -> f32
• weibull_cdf(x: f32, shape: f32, scale: f32) -> f32: 1 - exp(-(x/scale)^shape)
• weibull_inv_cdf(q: f32, shape: f32, scale: f32) -> f32

import Nautilus.Distributions (weibull_cdf)

cdf = weibull_cdf(cast(1.0, f32), cast(2.0, f32), cast(1.0, f32))

Beta

Parameterized by shape parameters a and b, both > 0. PDF is 0 outside [0, 1].

• beta_pdf(x: f32, a: f32, b: f32) -> f32: via log_gamma
• beta_cdf(x: f32, a: f32, b: f32) -> f32: via regularized incomplete beta

import Nautilus.Distributions (beta_cdf)

cdf = beta_cdf(cast(0.3, f32), cast(2.0, f32), cast(5.0, f32))  -- approximately 0.5798

F distribution

Parameterized by d1 and d2 degrees of freedom. Uses betai internally.

• f_pdf(x: f32, d1: f32, d2: f32) -> f32 / f_cdf(...) -> f32

import Nautilus.Distributions (f_cdf)

cdf = f_cdf(cast(3.0, f32), cast(5.0, f32), cast(10.0, f32))  -- approximately 0.918

Edge cases

Condition	Result
Any PDF with x < 0 (except Uniform)	0.0
exponential_inv_cdf(1.0, rate)	+inf
weibull_inv_cdf(0.0/1.0, ...)	0.0 / +inf
beta_pdf or f_pdf with params <= 0	NaN
lognormal_pdf(0.0, ...)	0.0