Other continuous distributions

Six more continuous families. Each has a PDF and CDF; Exponential, LogNormal, Uniform, and Weibull also have an inverse CDF, and the first three have a sampler. Every function is f32.

import Nautilus.Distributions (exponential_cdf, exponential_inv_cdf, lognormal_cdf, uniform_cdf, weibull_cdf, weibull_inv_cdf, beta_cdf, f_cdf, f_pdf)

exp_cdf = exponential_cdf(1.0f32, 2.0f32)
exp_median = exponential_inv_cdf(0.5f32, 2.0f32)
logn_cdf = lognormal_cdf(1.0f32, 0.0f32, 1.0f32)
unif_cdf = uniform_cdf(0.25f32, 0.0f32, 1.0f32)
weib_cdf = weibull_cdf(1.0f32, 2.0f32, 1.0f32)
weib_median = weibull_inv_cdf(0.5f32, 2.0f32, 1.0f32)
beta = beta_cdf(0.3f32, 2.0f32, 5.0f32)
f_c = f_cdf(3.0f32, 5.0f32, 10.0f32)
f_p = f_pdf(1.0f32, 5.0f32, 10.0f32)

exp_cdf = 0.86466473
exp_median = 0.3465736
logn_cdf = 0.5
unif_cdf = 0.25
weib_cdf = 0.63212055
weib_median = 0.83255464
beta = 0.57982504
f_c = 0.93444246
f_p = 0.49547988

Beta and F reject non-positive parameters with NaN. The other four families do not check their parameters; each section says what an invalid value returns, so validate computed parameters before the call.

Exponential

Parameterized by rate (lambda), not scale: the mean is 1 / rate. rate must be positive and finite.

Function	Signature	Returns
exponential_pdf	(x: f32, rate: f32) -> f32	rate * exp(-rate * x) for x >= 0, else 0
exponential_cdf	(x: f32, rate: f32) -> f32	1 - exp(-rate * x) for x >= 0, else 0
exponential_inv_cdf	(q: f32, rate: f32) -> f32	-ln(1 - q) / rate; 0 at q = 0, +inf at q = 1, NaN outside [0, 1]
exponential_sample	[n](k: key, template: tensor[n, f32], rate: f32) -> tensor[n, f32]	n draws of -ln(u) / rate

With rate = 0 the PDF and CDF are 0 everywhere and exponential_inv_cdf returns +inf. A negative rate gives negative values: exponential_pdf(1, -1) is -2.7182817 and exponential_cdf(1, -1) is -1.7182819.

The CDF complements in f64, and for a rate * x in the half-open interval from 0 up to 1/16 it sums a Maclaurin series for 1 - exp(-rate * x). A negative rate * x uses the subtraction branch. For small positive x, exponential_cdf(1e-8, 1.0) is 1e-8 and exponential_cdf(1e-20, 1.0) is 1e-20. The precision guide states the measured range.

LogNormal

Parameterized by mu and sigma, the mean and standard deviation of ln(X), not of X. sigma must be positive and finite.

Function	Signature	Returns
lognormal_pdf	(x: f32, mu: f32, sigma: f32) -> f32	density of X; 0 for x <= 0
lognormal_cdf	(x: f32, mu: f32, sigma: f32) -> f32	normal_cdf(ln(x), mu, sigma); 0 for x <= 0
lognormal_inv_cdf	(q: f32, mu: f32, sigma: f32) -> f32	exp(normal_inv_cdf(q, mu, sigma))
lognormal_sample	[n](k: key, template: tensor[n, f32], mu: f32, sigma: f32) -> tensor[n, f32]	exp of normal_sample draws

sigma behaves as std does in the normal distribution: at 0 the PDF is NaN and the CDF is a step at exp(mu); a negative sigma gives a negative PDF and the upper tail from the CDF.

Uniform

Parameterized by the endpoints lo and hi, which must satisfy lo < hi.

Function	Signature	Returns
uniform_pdf	(x: f32, lo: f32, hi: f32) -> f32	1 / (hi - lo) on [lo, hi], else 0
uniform_cdf	(x: f32, lo: f32, hi: f32) -> f32	0 at or below lo, 1 at or above hi, (x - lo) / (hi - lo) between
uniform_inv_cdf	(q: f32, lo: f32, hi: f32) -> f32	lo + q * (hi - lo)
uniform_sample	[n](k: key, template: tensor[n, f32], lo: f32, hi: f32) -> tensor[n, f32]	n draws, each lo + (hi - lo) * u for a uniform u in [0, 1]

The endpoints are not checked:

• lo = hi: uniform_pdf(lo, lo, lo) is inf and uniform_cdf(lo, lo, lo) is 0.
• lo > hi: the PDF and CDF are 0 between the endpoints, and uniform_inv_cdf(0.25, 1, 0) is 0.75, counting down from lo.
• uniform_inv_cdf does not reject q outside [0, 1]; it extrapolates, so uniform_inv_cdf(1.5, 0, 1) is 1.5.

Weibull

Parameterized by shape (k) and scale (lambda), both positive. The CDF is the closed form 1 - exp(-(x/scale)^shape).

Function	Signature	Returns
weibull_pdf	(x: f32, shape: f32, scale: f32) -> f32	density; 0 for x < 0
weibull_cdf	(x: f32, shape: f32, scale: f32) -> f32	P(X <= x); 0 for x < 0
weibull_inv_cdf	(q: f32, shape: f32, scale: f32) -> f32	scale * (-ln(1 - q))^(1/shape); 0 at q = 0, +inf at q = 1, NaN outside [0, 1]

shape <= 0 or scale <= 0 returns NaN, except where the result is fixed before the parameters are read: x < 0 gives 0, and q = 0 or q = 1 gives 0 or +inf. At x = 0 the PDF is 0 for shape > 1, 1/scale for shape = 1, and +inf for shape < 1.

weibull_cdf forms both (x/scale)^shape and its complement in f64, and below a (x/scale)^shape of 1/16 it sums a Maclaurin series for 1 - exp(-t). With positive shape and scale, weibull_cdf(1e-8, 2.0, 1.0) is 1e-16. The precision guide states the measured range.

Beta

Parameterized by shape parameters a and b, both positive.

Function	Signature	Returns
beta_pdf	(x: f32, a: f32, b: f32) -> f32	density, computed in log space with log_gamma; 0 outside [0, 1]
beta_cdf	(x: f32, a: f32, b: f32) -> f32	regularized incomplete beta I_x(a, b); 0 for x <= 0, 1 for x >= 1

a <= 0 or b <= 0 returns NaN. At x = 0 the PDF is 0 for a > 1, b for a = 1, and +inf for a < 1; x = 1 mirrors this with b.

The incomplete beta is computed in f64 and returned as f32. At the measured cases with a and b from 1 through 1e8, relative error was below 2e-6. These samples do not establish a bound between their points. The precision guide states the sampled axes and observed failures outside those limits.

F distribution

Parameterized by the numerator and denominator degrees of freedom d1 and d2, both positive. The CDF evaluates the regularized incomplete beta function at d1*x / (d1*x + d2).

Function	Signature	Returns
f_pdf	(x: f32, d1: f32, d2: f32) -> f32	density; 0 for x <= 0
f_cdf	(x: f32, d1: f32, d2: f32) -> f32	P(X <= x); 0 for x <= 0

d1 <= 0 or d2 <= 0 returns NaN for x > 0. Non-integer degrees of freedom are accepted.

Both d1*x / (d1*x + d2) and its complement d2 / (d1*x + d2) are formed in f64, so a small d2 beside a large d1*x keeps its digits instead of rounding the argument to 1. At the measured cases with d1 and d2 from 1 through 1e8, relative error was below 2e-6; this does not establish a bound between samples. Observed cases beyond those limits include f_cdf(0.5, 2e8, 0.5), with about 2e-6 relative error, and f_cdf(0.5, 1e8, 1e-3), with about 2.5e-5. The precision guide states the sampled axes for each export in this family.

Pitfalls

• Where the text above gives a CDF as 1 - exp(..), that is its definition and not how it is evaluated. Subtracting that way loses every small value, so both functions complement in f64 through a series instead. Their own small values are sound; what still loses everything below about 6e-8 is a caller writing 1 - exponential_cdf(x, rate) for the upper tail.
• Only the gamma family has survival functions. For the upper tail of these families, subtracting the CDF from 1 loses everything below about 6e-8.