Binomial distribution

The binomial distribution counts the number of successes when repeating

Bernoulli trials with success probability , assuming independent identically distributed trials.

Characteristics

Characteristic	Formula	Description
Notation		Sometimes or
Parameters		trials; success probability;
Support		All integers from 0 to
Probability mass function		For ; binomial coefficient
Cumulative distribution function		Also (regularized beta function)
Moment-generating function		Product of i.i.d. Bernoulli MGFs
Characteristic function		Fourier transform of the law
Mean		Average number of successes in trials
Variance		Grows with ; maximized at for fixed
Standard deviation		Natural fluctuation scale around
Mode		Two adjacent modes if is an integer
Median	or	Always lies between these two values
Skewness		Tends to 0 as (Gaussian regime)
Excess kurtosis		Decays as ; tails thin at large scale
Shannon entropy	—	No simple closed form; numerical computation via the PMF
Fisher information		For parameter from one observation
Kullback-Leibler divergence		Same number of trials ; reduces to Bernoulli divergence

Property / theorem	Formula	Description
Sum of Bernoulli variables		Canonical construction from Bernoulli
Case		Bernoulli is the elementary case
Additivity (same )	indep.	Stable under sums of independent trials
Complement of successes		Number of failures; success/failure symmetry
Factorial moments		For ; useful for variance and covariance
Normal approximation		De Moivre-Laplace / CLT; see normal distribution
Continuity correction		Improves Gaussian approximation for moderate
Poisson limit		Rare-event regime; see Poisson
Chernoff bound		Exponential tail bound; useful in ML and finance
Sampling without replacement	—	If trials are not independent (finite urn), the law is hypergeometric, not binomial

Number of defaults among homogeneous credits (common PD): baseline portfolio risk model under independence.

Number of defective items in a batch of size , under a constant defect rate assumption.

Number of conversions out of exposures; comparing two proportions across cohorts.

Multi-period modeling of an underlying (Cox-Ross-Rubinstein): each branch is a Bernoulli trial; the number of up moves is binomial.

Concept	Formula	Description
Observation	, observe successes	Equivalent to Bernoulli trials with successes
MLE estimator		Empirical proportion; unbiased for , natural estimator
Estimator variance		Same formula as for observed Bernoulli trials
Exact binomial test		Exact p-value under ; no normal approximation required
Proportion Z-test		Asymptotic approximation; valid when and are large
Clopper-Pearson interval		Exact confidence interval for
Wilson interval		Better behavior than Wald for near 0 or 1
Conjugate prior (Bayesian)		Posterior after successes in trials

Inference on from an observed matches that from Bernoulli trials — see Bernoulli.
Prefer the exact test for small or extreme ; the normal approximation simplifies large samples.
Comparing two groups ( vs ) uses a difference-of-proportions test or a independence test.
Rule of thumb: and before using the normal approximation.

Method	Description	Applications
Pseudo-random generation	Sum of Bernoulli trials or equivalent algorithm (inversion, BTPE)	Portfolio simulation, stress tests
NumPy	`np.random.binomial(n, p)` or `rng.binomial(n, p, size=m)`	independent draws from
SciPy	`scipy.stats.binom(n, p)` — PMF, CDF, `ppf`, `rvs`, moments	Exact probabilities, quantiles, simulation
Exact test	`scipy.stats.binomtest(k, n, p)`	Exact p-value for a proportion test
Normal approximation	`scipy.stats.norm.cdf` with continuity correction	Cumulative probabilities for large
Confidence intervals	`proportion_confint` (statsmodels) — Wilson, Clopper-Pearson	Robust estimation of

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