Negative binomial distribution

On the failures-before-the-

r

th-success convention, a negative binomial variable with success probability

p

has probability mass function

P (X = x) = (x x + r - 1) (1 - p)^{x} p^{r}, x = 0, 1, \dots .

It has expected value

r (1 - p) / p

r (1 - p) / p^{2}

.

Table of contents
- Negative binomial stopping argument Negative binomial distribution
- Poisson-gamma mixture Negative binomial distribution

Negative binomial stopping argument

For the

r

th success to occur after exactly

x

failures, the final trial must be a success, while the preceding

x + r - 1

trials contain

x

failures and

r - 1

successes. Choosing the failure locations gives the factor

(x x + r - 1)

.

Poisson-gamma mixture

If

Y ∣ Λ \sim Poisson (Λ)

and

Λ

has a gamma distribution, then

Y

has a negative binomial distribution. For gamma shape

ν

and scale

α

,

E Y = αν, Var (Y) = αν + α^{2} ν .

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Negative binomial distribution by

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The Negative Binomial distribution is a discrete probability distribution that models the number of trials needed to achieve a fixed number of successful outcomes (often referred to as "successes"). It is commonly used in scenarios where we are interested in the number of failures that occur before a certain number of successes is achieved. ### Key Characteristics: 1. **Parameters**: The Negative Binomial distribution is defined by two parameters: - \( r \): the number of successes (a positive integer).

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