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Binomial Distribution

NIST/SEMATECH Section 1.3.6.6.18 Binomial Distribution

PMF

Bin(n=10, p=0.5) PDF 0 2 4 6 8 10 k 0 0.05 0.1 0.15 0.2 0.25 P(X=k)

CDF

Bin(n=10, p=0.5) CDF 0 2 4 6 8 10 k 0 0.2 0.4 0.6 0.8 1 F(x)

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PMF

CDF

Formulas

Probability Mass Function

P(X=k)=(nk)pk(1−p)n−k,k=0,1,…,nP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, \ldots, n

Cumulative Distribution Function

F(k)=∑i=0⌊k⌋(ni)pi(1−p)n−iF(k) = \sum_{i=0}^{\lfloor k \rfloor} \binom{n}{i} p^i (1-p)^{n-i}

Properties

Mean

npnp

Variance

np(1−p)np(1-p)

Overview

The binomial distribution models the number of successes in nn independent Bernoulli trials, each with success probability pp. It is the foundation for binary outcome analysis and quality control sampling.

Related Distributions

Poisson Distribution Normal Distribution Beta Distribution