Bernoulli Process as Negative Binomial Distribution

Theorem
Let $$\left \langle{X_i}\right \rangle$$ be a Bernoulli process with parameter $p$.

Let $$\mathcal E$$ be the experiment which consists of performing the Bernoulli trial $$X_i$$ as many times as it takes to achieve a total of $$n$$ successes, and then stops.

Let $$X$$ be the discrete random variable defining the number of trials before $$n$$ successes have been achieved.

Then $$X$$ is modelled by a negative binomial distribution $$\operatorname{NB} \left({n, p}\right)$$.

Proof
First note that the number of Bernoulli trials has to be at least $$n$$, so the image is correct: $$\operatorname{Im} \left({X}\right) = \left\{{n, n+1, n+2, \ldots}\right\}$$.

Now, note that if $$X$$ takes the value $$x$$, then in the first $$x-1$$ trials there must have been $$n-1$$ successes.

Hence there must have been $$x-n$$ failures, and so a success happens at trial number $$x$$.

So the probability of the occurrence of the event $$\left[{X = x}\right]$$ is given by the binomial distribution, as follows:
 * $$p_X \left({x}\right) = \binom {x - 1} {n - 1} p^n \left({1-p}\right)^{x - n}$$

where $$x \in \left\{{k, k+1, k+2, \ldots}\right\}$$

Hence the result, by definition of negative binomial distribution.