Definition:Negative Binomial Distribution/Second Form
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Definition
Let $X$ be a discrete random variable on a probability space $\left({\Omega, \Sigma, \Pr}\right)$.
$X$ has the negative binomial distribution (of the second form) with parameters $n$ and $p$ if:
- $\Img X = \set {n, n + 1, n + 2, \dotsc}$
- $\map \Pr {X = k} = \dbinom {k - 1} {n - 1} p^n \paren {1 - p}^{k - n}$
where $0 < p < 1$.
It is frequently seen as:
- $\map \Pr {X = k} = \dbinom {k - 1} {n - 1} q^{k - n} p^n $
where $q = 1 - p$.
Notation
The negative binomial distribution (in either form) can be written:
- $X \sim \NegativeBinomial n p$
but there is no standard notation for this distribution.
Also see
- Bernoulli Process as Negative Binomial Distribution/Second Form
- Negative Binomial Distribution (Second Form) as Generalized Geometric Distribution
- Negative Binomial Distribution (Second Form) Gives Rise to Probability Mass Function
- Results about the negative binomial distribution can be found here.
Technical Note
The $\LaTeX$ code for \(\NegativeBinomial {n} {p}\) is \NegativeBinomial {n} {p}
.
When the arguments are single characters, it is usual to omit the braces:
\NegativeBinomial n p
Sources
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 2.2$: Examples: $(10)$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): negative binomial distribution
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $13$: Probability distributions
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $15$: Probability distributions