PLAYGROUND

DISTRIBUTIONS / DISCRETE / BINOMIAL

Binomial distribution

The idea in one sentence

The Binomial distribution counts successes in n independent Bernoulli trials with common probability p. It is a count model with a fixed number of opportunities.

Support and interpretation

Support is {0,1,…,n}. n sets the maximum and p moves mass between zero and n; relative variance is largest near p=1/2.

Its place in the family

It is a Bernoulli sum and approaches Poisson when n grows, p is small, and np stays moderate. For large n it can also be approximated by Normal with continuity correction.

An applied reading

The Binomial question is “how many successes in these n opportunities?”. If opportunities are random, Poisson, Negative Binomial, or a hierarchical model may fit the process better.

Do not use Binomial merely because the response is 0/1; simple Binomial requires fixed trial count and common probability.

The denominator is part of the story

A count of 18 positive responses says little until we know whether there were 20 opportunities or 20,000. In a Binomial model, n is not an administrative detail: it sets the largest possible count and determines its possible variation. This is why the model fits inspected batches, repeated throws, and surveys with a known number of responses.

The Poisson approximation is useful when opportunities are numerous and each success is rare, provided np stays at a moderate scale. The Normal approximation works when enough probability remains away from both endpoints. Neither approximation repairs dependence, contagion between trials, or changing probabilities. If observed variation clearly exceeds np × [1-p], investigate heterogeneity, clustering, or a beta-binomial model.

Decision guide

A good candidate when: successes are counted across a fixed number of independent trials sharing one probability.

Compare it with: Poisson for rare events and Beta-binomial under overdispersion or heterogeneous probabilities. Verify that n is genuinely fixed and known.

References

Binomial Distribution: equations and calculator

Distribution defintion

X∼Binomial(n,p)X\sim\mathrm{Binomial}\left(n,p\right)

Distribution domain

x∈N≡{0,1,2,… }x\in\mathbb{N}\equiv \left\{ 0,1,2,\dots \right\}

Parameters domain and parameters constraints

n∈N,p∈(0,1)⊆Rn\in\mathbb{N},p\in\left(0,1\right)\subseteq\mathbb{R}

Cumulative distribution function

FX(x)=∑i=0x(ni)pi(1−p)n−i=I(1−p,n−x,1+x)F_{X}\left(x\right)=\sum_{i=0}^{x} \binom{n}{i} p^i(1-p)^{n-i}=I(1-p, n - x, 1 + x)

Probability mass function

fX(x)=(nx)px(1−p)n−xf_{X}\left(x\right)=\binom{n}{x} p^x (1-p)^{n-x}

Percent point function/Sample

FX−1(u)=arg⁡min⁡x∣FX(x)−u∣F^{-1}_{X}\left(u\right)=\arg\min_{x}\left| F_{X}\left(x\right)-u \right|

Parametric centered moments

E[Xk]=μk′=∑x=0∞xkfX(x)=∑i=0kn!(n−i)!S(k,i)piE[X^k]=\mu'_{k}=\sum_{x=0}^{\infty }x^{k}f_{X}\left(x\right)=\sum_{i=0}^k\tfrac{n!}{(n-i)!}S(k,i)p^{i}

Parametric mean

Mean(X)=μ1′=np\mathrm{Mean}(X)=\mu'_{1}=np

Parametric variance

Variance(X)=(μ2′−μ1′2)=np(1−p)\mathrm{Variance}(X)=(\mu'_{2}-\mu'^{2}_{1})=np(1-p)

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=1−2pnp(1−p)\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{1-2p}{\sqrt{np(1-p)}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3+1−6p(1−p)np(1−p)\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=3+\frac{1-6p(1-p)}{np(1-p)}

Parametric median

Median(X)=⌊np⌋∨⌈np⌉\mathrm{Median}(X)=\lfloor{np}\rfloor \vee \lceil{np}\rceil

Parametric mode

Mode(X)=⌊(n+1)p⌋∨⌈(n+1)p⌉−1\mathrm{Mode}(X)=\lfloor (n + 1)p \rfloor \vee \lceil (n + 1)p \rceil - 1

Additional information and definitions

Computing an analytic expression for the inverse of the cumulative distribution function is not feasible. However, it is possible to calculate the Percentile Point Function by approximating it to the nearest integer.
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
⌊x⌋:Floor function\lfloor{x}\rfloor: \text{Floor function}
⌈x⌉:Ceiling Function\lceil{x}\rceil: \text{Ceiling Function}
I(x,a,b):Regularized incomplete beta functionI\left(x,a,b\right):\text{Regularized incomplete beta function}
S(a,b):Stirling numbers of the second kind=1b!∑j=0b(−1)b−j(bj)jaS(a,b):\text{Stirling numbers of the second kind}=\frac1{b!}\sum_{j=0}^b(-1)^{b-j}\binom {b}{j} j^a