torch.distributions.Binomial

class Binomial extends Distribution

new Binomial(options: {
total_count: number | Tensor;
probs?: number | Tensor;
logits?: number | Tensor;
} & DistributionOptions)

readonlytotal_count(Tensor): – Number of Bernoulli trials.
readonlyarg_constraints(unknown)
readonlysupport(unknown)
readonlyprobs(Tensor)
readonlylogits(Tensor)
readonlymean(Tensor)
readonlymode(Tensor)
readonlyvariance(Tensor)

Binomial distribution: discrete distribution for counting successes.

Number of successes in n independent Bernoulli trials with probability p. Essential for:

Counting successes/failures in fixed number of trials
Bernoulli process outcomes (coin flips, dice rolls)
Quality control and defect counting
A/B testing and conversion counting
Hypothesis testing (exact binomial test)
Survey/polling data (number of yes responses)
Bayesian inference with Beta priors (conjugate)
Approximation: Normal for large n (central limit theorem)

The probability mass function: P(X = k) = C(n, k) * p^k * (1-p)^(n-k) where C(n, k) = n! / (k! * (n-k)!)

\begin{aligned} \text{PMF: } P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \quad k = 0,1,\ldots,n \\ \text{Mean: } \mathbb{E}[X] = np \\ \text{Variance: } \text{Var}(X) = np(1-p) \\ \text{Mode: } \lfloor (n+1)p \rfloor \\ \text{Entropy: } H(X) = \text{log}(2\pi np(1-p))/2 + O(1/n) \end{aligned}

Discrete support: Samples are non-negative integers 0 to n
Symmetric: For p=0.5, distribution is symmetric around n/2
Conjugate prior: Beta distribution is conjugate to Binomial
Sum property: Sum of independent binomials with same p is binomial
Limiting distributions: Normal (large n), Poisson (large n, small np)
Central limit theorem: Standardized → Normal as n → ∞

Parameter constraints: n must be non-negative integer, p ∈ [0, 1]
Numerical overflow: Very large n or extreme p can overflow
Approximations: Normal approximation needs np ≥ 5 and n(1-p) ≥ 5
Probability range: Binomial(n, 0) = 0, Binomial(n, 1) = n always

Examples

// Basic binomial: 100 trials with 30% success rate
const m = new torch.distributions.Binomial({ total_count: 100, probs: 0.3 });
m.sample();  // Number of successes in 100 trials

// Coin flips: Bernoulli as special case (n=1)
const coin = new torch.distributions.Binomial({ total_count: 1, probs: 0.5 });
coin.sample();  // 0 or 1 (heads/tails)

// Multiple coin flips: 10 trials
const flips = new torch.distributions.Binomial({ total_count: 10, probs: 0.5 });
flips.sample();  // Number of heads in 10 flips

// A/B testing: count successes in treatment group
const num_trials = 1000;
const conversion_rate = 0.05;  // 5% conversion
const conversions = new torch.distributions.Binomial({
  total_count: num_trials,
  probs: conversion_rate
});
const obs_conversions = conversions.sample();  // Observed count

// Batched: different probabilities
const counts = torch.tensor([10, 20, 30]);  // Different trial counts
const probs = torch.tensor([0.3, 0.5, 0.7]);  // Different probabilities
const dist = new torch.distributions.Binomial({
  total_count: counts,
  probs: probs
});
const samples = dist.sample();  // Success count for each

// Bayesian inference: using Beta conjugate prior
// Prior: Beta(α, β) represents prior belief about p
// Likelihood: Binomial(n, p) observes k successes
// Posterior: Beta(α+k, β+n-k)
const num_successes = 30;
const num_failures = 70;
const n = num_successes + num_failures;

const prior_alpha = 1, prior_beta = 1;  // Uninformative Beta(1,1)
const posterior_alpha = prior_alpha + num_successes;
const posterior_beta = prior_beta + num_failures;
// Posterior distribution over p

// Survey polling: count "yes" responses
const sample_size = 500;  // Survey n=500 people
const support_rate = 0.55;  // 55% support in population
const survey = new torch.distributions.Binomial({
  total_count: sample_size,
  probs: support_rate
});
const yes_votes = survey.sample();  // Expected ~275, but varies