Geometric Distribution

In the special case of the Negative Binomial where $r = 1$ , the negative binomial pmf simplifies to:

nb (x; 1, p) = (1 - p)^{x - 1} p : x = 1, 2, 3, \dots

The random variable $X =$ the number of trials required to achieve one success is referred to as a geometric random variable, with the pmf as the geometric distribution. The idea is that the probabilities are a geometric progression:

p, (1 - p) p, (1 - p)^{2} p, . . .

The sum of these probabilities rests on the fact that:

a + a r + a r^{2} + \dots = \frac{a}{1 - r} : | r | < 1

So then:

\begin{aligned} P (X \leq x) & = \sum_{n = 1}^{x} p (1 - p)^{n - 1} \\ = p \sum_{n = 1}^{x} (1 - p)^{n - 1} \\ = \frac{p}{1 - p} \cdot \frac{1}{1 - (1 - p)} \\ = \frac{p}{p (1 - p)} \\ = \frac{1}{1 - p} \end{aligned}