Discrete random variable

Negative binomial distribution

The negative binomial distribution 𝑋 ∼NBin(𝑟,𝑝) describes the number of failures of independent Bernoulli trials with success probability 𝑝 before the 𝑟-th success. prob It is thus the sum of 𝑟 independent geometrically distributed variables, whence follows the probability mass function

ℙ(𝑋=𝑥)=𝑝𝑟(𝑥+𝑟−1𝑟−1)(1−𝑝)𝑥

Proof by induction

In the case 𝑋 ∼NBin(1,𝑝) it reduces to the Geometric distribution 𝑋 ∼Geom(𝑝). Now assume the probability mass function above is valid for 𝑋𝑟 ∼NBin(𝑟,𝑝). Let 𝑌 ∼Geom(𝑝) be independent so that 𝑋𝑟+1 =𝑋𝑟 +𝑌 ∼NBin(𝑟 +1,𝑝). Then

ℙ(𝑋𝑟+1=𝑥)=ℙ(𝑋𝑟+𝑌=𝑥)=𝑥∑𝑖=0ℙ(𝑋𝑟+𝑌=𝑥∣𝑋𝑟=𝑖)ℙ(𝑋𝑟=𝑖)=𝑥∑𝑖=0ℙ(𝑌=𝑥−𝑖)ℙ(𝑋𝑟=𝑖)=𝑥∑𝑖=0(1−𝑝)𝑥−𝑖𝑝𝑝𝑟(𝑖+𝑟−1𝑟−1)(1−𝑝)𝑖=𝑝𝑟+1(1−𝑝)𝑥𝑥∑𝑖=0(𝑖+𝑟−1𝑟−1)=𝑝𝑟+1(𝑥+𝑟𝑟)(1−𝑝)𝑥

where on the final line we invoked ^P5.

Properties

Let 𝑋 ∼NBin(𝑟,𝑝) and 𝑞 =1 −𝑝

1. Expectation: 𝜇 =𝔼⁡[𝑋] =𝑟𝑞𝑝
2. Variance: 𝜎2 =Var⁡[𝑋] =𝑟𝑞𝑝2
3. Moment-generating function: 𝑀𝑋(𝑡) =(𝑝1−𝑞e𝑡)𝑟 for 𝑞e𝑡 <1
4. Probability-generating function: 𝑔𝑋(𝑡) =(𝑝1−𝑞𝑡)𝑟

Proof of 1–3

^P1 follows from ^P1 by linearity of expectation, while ^P2 follows from ^P2 by ^P3 ^P3 follows from ^P3 by ^P1.

Relationship to other distributions

• By the Central limits theorem, NBin(𝑛,𝑝) ⇝N⁡(𝑛𝑞𝑝,𝑛𝑞𝑝2) as 𝑛 →∞

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