negative binomial distribution

Negative binomial properties:

1. A series of trails, classed as being either ‘success‘ or ‘failure‘ (Bernoulli trial).

2. The trials are observed until exactly r successes are obtained, where r is fixed by the experimenter.

3. The random variable $X$ denotes the number of successes needed to obtain the r trials.

E.g:

Sample space for an experiment in which $r = 3$:

$$S = { ssffffs, sffffss, ffffsss, sss, ssfs }$$

Here, $X = 7, 7, 7, 3, 4$, for $X = 7$ , there are $left( begin{array}{c}{ x-1 } \ { 2 } end{array} right)$ ways in which $X$ can assume the value 7.

The probability of an outcome for which $X = x$ is given by:

$$P[X = x] = left( begin{array}{c}{ x-1 } \ { 2 } end{array} right)(1-p)^x-3p^3 qquad x=3, 4, 5...$$

Definition ( Negative binomial distribution):

$$f(x) = left( begin{array}{l}{ x-1 } \ { 2-1 } end{array} right) p^r(1-p)^{x-r} qquad r=1, 2, 3,... qquad x = r, r+1, r+2,...$$

Negative binomial moment generating function:

$$m_X(t) = dfrac{(pe^t)^r}{(1-qe^t)^r} qquad q = 1-p$$

Expectation and Variance for the Negative Binomial Distribution:

$$E[X] = mu = r/p qquad and qquad operatorname{Var} [X] = sigma^2 = rq/p^2$$