Discrete random variable

Binomial distribution

A binomial distribution is the sum of 𝑛 independent Bernoulli trials with probability of success 𝑝. prob

𝑌𝑖∼Bern⁡(𝑝)𝑋=𝑛∑𝑖=1𝑌𝑖𝑋∼Bin⁡(𝑛,𝑝)

The probability of a given value of 𝑋 =𝑥 ∈ℕ0 is hence given by

ℙ(𝑋=𝑥)=(𝑛𝑥)𝑝𝑥(1−𝑝)𝑛−𝑥

and the Expectation and variance are given by

𝔼⁡(𝑋)=𝑛𝑝Var⁡(𝑋)=𝑛𝑝(1−𝑝)

As 𝑛 →∞, the shape of a binomial distribution approaches that of the continuous Normal distribution. See binomial coëfficient.

Properties

Let 𝑋 ∼Bin⁡(𝑛,𝑝) and let 𝑞 =1 −𝑝

1. Expectation: 𝜇 =𝔼⁡[𝑋] =𝑛𝑝
2. Variance: 𝜎2 =Var⁡[𝑋] =𝑛𝑝𝑞
3. Moment-generating function: 𝑀𝑋 :ℝ →ℝ :𝑡 ↦(𝑝e𝑡 +𝑞)𝑛
4. Probability-generating function: 𝑔𝑋(𝑡) =(𝑝𝑡 +𝑞)𝑛

Proof of 1–4

These follow from the analogous results for a Bernoulli trial by ^P2, ^P3 and ^P1, since a binomial variable may be thought of as the sum of independent and identically distributed Bernoulli trials. ^P4 follows directly from the Binomial expansion.

Some further properties

1. 𝑛 −𝑋 ∼Bin⁡(𝑛,𝑞)
2. 𝑋 +𝑌 ∼Bin⁡(𝑛 +𝑚,𝑝) if 𝑌 ∼Bin⁡(𝑛 =𝑚,𝑝) is independent from 𝑋

Relationship to other distributions

• Relationship between binomial and hypergeometric distributions
• By the Central limits theorem, Bin⁡(𝑛,𝑝) ⇝N⁡(𝑛𝑝,𝑛𝑝𝑞) as 𝑛 →∞.

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