# Category:Negative Binomial Distribution

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This category contains results about the negative binomial distribution.

Definitions specific to this category can be found in Definitions/Negative Binomial Distribution.

Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

There are two forms of the **negative binomial distribution**, as follows:

### First Form

$X$ has the **negative binomial distribution (of the first form) with parameters $n$ and $p$** if:

- $\Img X = \set {0, 1, 2, \ldots}$

- $\map \Pr {X = k} = \dbinom {n + k - 1} {n - 1} p^k \paren {1 - p}^n$

where $0 < p < 1$.

It is frequently seen as:

- $\map \Pr {X = k} = \dbinom {n + k - 1} {n - 1} p^k q^n$

where $q = 1 - p$.

### Second Form

$X$ 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 $q = 1 - p$.

## Pages in category "Negative Binomial Distribution"

The following 31 pages are in this category, out of 31 total.

### B

### D

### E

### F

### N

- Negative Binomial Distribution (First Form) as Generalized Geometric Distribution
- Negative Binomial Distribution (First Form) Gives Rise to Probability Mass Function
- Negative Binomial Distribution (Second Form) as Generalized Geometric Distribution
- Negative Binomial Distribution (Second Form) Gives Rise to Probability Mass Function
- Negative Binomial Distribution as Generalized Geometric Distribution
- Negative Binomial Distribution as Generalized Geometric Distribution/First Form
- Negative Binomial Distribution as Generalized Geometric Distribution/Second Form
- Negative Binomial Distribution Gives Rise to Probability Mass Function
- Negative Binomial Distribution Gives Rise to Probability Mass Function/First Form
- Negative Binomial Distribution Gives Rise to Probability Mass Function/Second Form

### P

- Probability Generating Function of Negative Binomial Distribution
- Probability Generating Function of Negative Binomial Distribution (First Form)
- Probability Generating Function of Negative Binomial Distribution (Second Form)
- Probability Generating Function of Negative Binomial Distribution/First Form
- Probability Generating Function of Negative Binomial Distribution/Second Form