Discrete random variable

Negative binomial distribution

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

Proof by induction

In the case it reduces to the Geometric distribution . Now assume the probability mass function above is valid for . Let be independent so that . Then

where on the final line we invoked ^P5.

Properties

Let and

1. Expectation:
2. Variance:
3. Moment-generating function: for
4. Probability-generating function:

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, as

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