Bernoulli Process as Binomial Distribution

Theorem
Let $\left \langle{X_i}\right \rangle$ be a finite Bernoulli process of length $n$ such that each of the $X_i$ in the sequence is a Bernoulli trial with parameter $p$.

Then the number of successes in $\left \langle{X_i}\right \rangle$ is modelled by a binomial distribution with parameters $n$ and $p$.

Hence it can be seen that:
 * $X \sim \operatorname{B} \left({1, p}\right)$ is the same thing as $X \sim \operatorname{Bern} \left({p}\right)$

Proof
Consider the sample space $\Omega$ of all sequences $\left \langle{X_i}\right \rangle$ of length $n$.

The $i$th entry of any such sequence is the result of the $i$th trial.

We have that $\Omega$ is finite.

Let us take the event space $\Sigma$ to be the power set of $\Omega$.

As the elements of $\Omega$ are independent, by definition of the Bernoulli process, we have that:
 * $\forall \omega \in \Omega: \Pr \left({\omega}\right) = p^{s \left({\omega}\right)} \left({1 - p}\right)^{n - s \left({\omega}\right)}$

where $s \left({\omega}\right)$ is the number of successes in $\omega$.

In the same way:
 * $\displaystyle \forall A \in \Sigma: \Pr \left({A}\right) = \sum_{\omega \in A} \Pr \left({\omega}\right)$

Now, let us define the discrete random variable $Y_i$ as follows:
 * $Y_i \left({\omega}\right) = \begin{cases}

1 : & \omega_i \text { is a success} \\ 0 : & \omega_i \text { is a failure} \\ \end{cases}$ where $\omega_i$ is the $i$th element of $\omega$.

Thus, each $Y_i$ has image $\left\{{0, 1}\right\}$ and a probability mass function:
 * $\Pr \left({Y_i = 0}\right) = \Pr \left({\left\{{\omega \in \Omega: \omega_i \text { is a success}}\right\}}\right)$

Thus we have:

Then:
 * $\Pr \left({Y_i = 0}\right) = 1 - \Pr \left({Y_i = 1}\right) = 1 - p$

So (by a roundabout route) we have confirmed that $Y_i$ has the Bernoulli distribution with parameter $p$.

Now, let us define the random variable $\displaystyle S_n \left({\omega}\right) = \sum_{i=1}^n Y_i \left({\omega}\right)$.

It is clear that:
 * $S_n \left({\omega}\right)$ is the number of successes in $\omega$;
 * $S_n$ takes values in $\left\{{0, 1, 2, \ldots, n}\right\}$ (as each $Y_i$ can be $0$ or $1$).

Also, we have that:

Hence the result.