Binomial Distribution Calculator

The binomial distribution is one of the most mathematically tractable discrete distributions, with closed-form expressions for all moments, a simple generating function, and well-characterized limit theorems. The binomial distribution calculator for mathematics courses provides both the numerical results and the mathematical machinery behind them.

The PMF: Combinatorial Derivation

The binomial PMF follows from first principles. Consider n independent Bernoulli trials, each with success probability p. A specific sequence of k successes and (n−k) failures has probability p^k × (1−p)^(n−k). The number of such sequences is the binomial coefficient C(n,k) = n! / (k!(n−k)!). Therefore:

P(X = k) = C(n,k) × p^k × (1−p)^(n−k), for k = 0, 1, ..., n

Verification that probabilities sum to 1: sum over all k of C(n,k) × p^k × (1−p)^(n−k) = (p + (1−p))^n = 1^n = 1, by the binomial theorem — the reason this distribution carries the name "binomial." Use this online calculator for any parameters. The applied binomial calculator covers practical applications.

Moment Generating Function and Moments

The moment generating function (MGF) of Binomial(n, p):

M_X(t) = E[e^(tX)] = (1 − p + p × e^t)^n = (q + p × e^t)^n where q = 1−p

All moments are derived by differentiating the MGF:

• Mean: E[X] = M'(0) = n × p
• E[X²] = M''(0) = n × p × (1 − p) + (np)² = npq + n²p²
• Variance: Var(X) = E[X²] − (E[X])² = npq = n × p × (1−p)
• Skewness: (1 − 2p) / √(npq)
• Excess kurtosis: (1 − 6pq) / (npq)

The probability generating function G_X(z) = (q + pz)^n is equally useful for computing factorial moments.

Limit Theorems: Poisson and Normal Approximations

Two fundamental limit theorems characterize the binomial's large-n behavior:

• Poisson limit (Law of Rare Events): as n → ∞ and p → 0 with np = λ constant: Binomial(n,p) → Poisson(λ). Proof: C(n,k) × p^k × (1−p)^(n−k) → λ^k × e^(−λ) / k! as n→∞. Applicable when n ≥ 20 and p ≤ 0.05.
• Normal approximation (Central Limit Theorem): as n → ∞ with p fixed, (X − np)/√(npq) → N(0,1) in distribution. The de Moivre-Laplace theorem proves this for the binomial specifically; CLT generalizes it to all finite-variance distributions. Practically valid when np ≥ 5 and nq ≥ 5.

The normal distribution calculator and probability distribution calculators cover these limiting distributions.

Sums and Convolution of Binomial Variables

If X ~ Binomial(n, p) and Y ~ Binomial(m, p) independently, then X + Y ~ Binomial(n+m, p). This reproductive property follows from the MGF: M_{X+Y}(t) = M_X(t) × M_Y(t) = (q+pe^t)^n × (q+pe^t)^m = (q+pe^t)^(n+m). The condition that both have the same p is essential — sums of binomials with different p values do not follow a binomial distribution. This convolution property makes the binomial distribution useful for combining independent identical experiments: if 5 batches of 100 items each have defect probability 0.03, the total defects across all batches follows Binomial(500, 0.03).