Bernoulli Process as Negative Binomial Distribution

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

First Form
Let $$\mathcal E$$ be the experiment which consists of performing the Bernoulli trial $$X_i$$ until a total of $$n$$ failures have been encountered.

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

Then $$X$$ is modelled by a negative binomial distribution of the first form.

Second Form
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 $$Y$$ 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 of the first form.

Proof for First Form
The number of Bernoulli trials may be as few as $$0$$, so the image is correct: $$\operatorname{Im} \left({X}\right) = \left\{{0, 1, 2, \ldots}\right\}$$.

If $$X$$ takes the value $$x$$, then there must have been $$n + x$$ trials altogether.

So, after $$n + x - 1$$ trials, there must have been $$n - 1$$ failures, as (from the description of the experiment) the last trial is a failure.

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

where $$x \in \left\{{0, 1, 2, \ldots}\right\}$$

Hence the result, by definition of first form of the negative binomial distribution.

Proof for Second Form
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:
 * $$p_X \left({x}\right) = \binom {x - 1} {n - 1} \left({1-p}\right)^{x - n} p^n$$

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

Hence the result, by definition of second form of the negative binomial distribution.