Bernoulli Process as Geometric 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 success, and then stop.

Let $$k$$ be the number of failures before a success is achieved.

Then $$k$$ is modelled by a geometric distribution with parameter $p$.

Proof
Follows directly from the definition of geometric distribution.

Let $$Y$$ be the discrete random variable defined as the number of failures before a success is achieved.

The probability that $$k$$ failures are followed by a success is:
 * $$P \left({Y = k}\right) = \left({1 - p}\right)^k p$$

Hence the result.