Discrete distributions (Poisson, Binomial)

Two discrete distribution families: Poisson and Binomial. Both provide a probability mass function (PMF) and a cumulative distribution function (CDF). The count parameter k is passed as f32 but must be integer-valued; non-integer k returns 0.

Poisson

poisson_pmf(k: f32, lambda: f32) -> f32

Computes P(X = k) = exp(k*ln(lambda) - lambda - ln(Gamma(k+1))) in log-space to avoid factorial overflow. Returns 0 for non-integer or negative k. At lambda = 0, returns 1 if k = 0, else 0.

poisson_cdf(k: f32, lambda: f32) -> f32

Computes P(X <= k) via the complement of the regularized lower incomplete gamma function: 1 - gammaP(k+1, lambda). This avoids summing individual PMF values.

Parameters: k >= 0 (integer-valued f32), lambda > 0
lambda < 0: returns NaN
lambda = 0: CDF returns 1.0 for all k >= 0

import Nautilus.Distributions (poisson_pmf, poisson_cdf)

pmf = poisson_pmf(cast(3.0, f32), cast(2.5, f32))  -- approximately 0.2138
cdf = poisson_cdf(cast(3.0, f32), cast(2.5, f32))  -- approximately 0.7576

Binomial

binomial_pmf(k: f32, n: f32, p: f32) -> f32

Computes P(X = k) = C(n,k) * p^k * (1-p)^(n-k) in log-space using log_gamma for the binomial coefficient. Returns 0 for non-integer k or k outside [0, n].

binomial_cdf(k: f32, n: f32, p: f32) -> f32

Computes P(X <= k) via the regularized incomplete beta function: betaI(n-k, k+1, 1-p). Returns 0 for k < 0, returns 1 for k >= n.

Parameters: k in [0, n] (integer-valued f32), n positive integer, p in [0, 1]
p = 0: returns 1 if k = 0, else 0
p = 1: returns 1 if k = n, else 0
p outside [0, 1]: returns NaN

import Nautilus.Distributions (binomial_pmf, binomial_cdf)

pmf = binomial_pmf(cast(3.0, f32), cast(10.0, f32), cast(0.3, f32))
-- approximately 0.2668
cdf = binomial_cdf(cast(3.0, f32), cast(10.0, f32), cast(0.3, f32))
-- approximately 0.6496

Edge cases

Condition	PMF	CDF
Non-integer k	0.0	uses floor(k) implicitly
k < 0	0.0	0.0
lambda < 0 (Poisson)	NaN	NaN
p outside [0,1] (Binomial)	NaN	NaN
k > n (Binomial)	0.0	1.0