Negative Binomial Distribution¶

Suppose we run an experiment with independent Bernoulli trials where the experiment stops when \(r > 0\) successes are observed. Let \(p\) be the probability of success, and \(k\) be the number of failures in the experiment,

\[ \begin{aligned} P(X = k) &= \binom{k + r - 1}{r-1}(1-p)^{k}p^{r} \quad k = 0, 1, 2, \ldots \end{aligned} \]

since the last trial is by definition a success; we can only choose the failures from the remaining trials. Again, \(X\) is the number of failures and not the number of trials.

Moment Generating Function¶

\[ \begin{aligned} \phi(t) &= \sum_{k=0}^{\infty}e^{tk} \binom{k+r-1}{r-1}\roundbr{1-p}^{k}p^{r}\newline \phi(t) &= \sum_{k=0}^{\infty} \binom{k+r-1}{ r-1}\squarebr{\roundbr{1-p}e^{t}}^{k}p^{r}\newline \phi(t) &= \sum_{k=0}^{\infty} \binom{k+r-1}{r-1}\squarebr{\roundbr{1-p}e^{t}}^{k}\squarebr{1 - \roundbr{1-p}e^{t}}^{r} \roundbr{\frac{p}{1 - \roundbr{1-p}e^{t}}}^{r}\newline \phi(t) &= \roundbr{\frac{p}{1 - \roundbr{1-p}e^{t}}}^{r} \sum_{k=0}^{\infty} \binom{k+r-1}{r-1}\squarebr{\roundbr{1-p}e^{t}}^{k}\squarebr{1 - \roundbr{1-p}e^{t}}^{r}\newline &= \roundbr{\frac{p}{1 - \roundbr{1-p}e^{t}}}^{r} \times 1\newline &= \roundbr{\frac{p}{1 - \roundbr{1-p}e^{t}}}^{r} \end{aligned} \]

Mean and Variance¶

With the moment generating function, mean and variance are easy to calculate

\[ \begin{aligned} E[X] &= \frac{r(1-p)}{p}\newline Var(X) &= \frac{r(1-p)}{p^{2}} \end{aligned} \]

Relation to Geometric Distribution¶

Geometric distribution is a special case of Negative binomial distribution with \(r = 1\)

\[ \begin{aligned} Geom(p) = NB(1, p) \end{aligned} \]

and can be checked using the mgf of the two.

Further, the sum of \(r\) independent geometric random variables is a negative binomial distribution with parameters \(r\) and \(p\)

\[ \begin{aligned} \sum_{r} Geom(p) = NB(r, p) \end{aligned} \]

Sum of Negative Binomial Random Variables¶

If \(X_{i}\) are \(NB(r_{i}, p)\), then the sum of \(n\) such variables is \(NB(\sum r_{i}, p)\). That is, the sum of negative binomial random variables is also a negative binomial random variable. This can be readily seen from the mgf.