Negative Binomial Distribution

The negative binomial distribution models the probability distribution of the number of independent Bernoulli trials required to observe a predetermined number of failures, where each trial has only two possible outcomes (success or failure) and the probability of success remains constant across all trials.

A negative binomial random variable is denoted as followed:

\[ X \sim \text{Negative Binomial}(r, p) \]

where

• \(r\) = number of sucessess

• \(p\) = probability of success in a Bernoulli trial

Probability Mass Function (PMF)

The pmf of the negative binomial distribution is as follows:

\[\begin{split} p_X(x) = \begin{cases} \binom{x-1}{r-1}p^r(1-p)^{x-r}, & \text{if } x=r, r+1, r+2, \dots \\ 0, & \text{otherwise} \end{cases} \end{split}\]

Expectation

To derive the expected value of the negative binomial distribution, we must first note the following:

\[ X = W_1 + W_2 + \dots + W_r = \sum_{i=1}^rW_i \]

where

\[ W_1, W_2, \dots, W_r \sim \text{Geometric}(p) \]

Because we know that the expected value of the geometric distribution is \(\frac{1}{p}\), we can use this to evaluate the expected value of the negative binomial distribution:

\[\begin{split} \begin{align} EX &= E \left( \sum_{i=1}^r W_i \right) \\ &= \frac{r}{p} \end{align} \end{split}\]

Variance

We can similarly evaluate the variance of the negative binomial distribution:

\[ Var(X) = \frac{r(1-p)}{p^2} \]