**The Bernoulli distribution is a theoretical model used to represent a discrete random variable which can only end in two mutually exclusive results. **

Recommended articles: **Bernoulli distribution, Bernoulli** example, **sample space** and Laplace Rule.

## Bernoulli probability function

Probability distribution function of the Bernoulli distribution.

We define **z** as the random variable Z once known and fixed. That is, Z is changing randomly (the dice rotates and spins in a single throw) but when we observe it we set the value (when the dice falls on the table and gives a concrete result). It is at that moment when we evaluate the result and assign one (1) or zero (0) depending on what we consider “success” or not “success”.

The random variable Z once set can only take two specific values: zero (0) or one (1). Then, the probability distribution function of the Bernoulli distribution will only be non-zero (0) when z is zero (0) or one (1). The opposite would be that the Bernoulli distribution distribution function was zero (0) since z will be any non-zero value (0) or one (1).

The previous function can also be rewritten as:

Probability distribution function of the Bernoulli distribution.

If we substitute z = 1 in the first formula of the probability function we will see that the result is p that coincides with the value of the second probability function when z = 1. Similarly, when z = 0 we get (1-p) for any value of p.

## Moments of the function

The moments of a distribution function are specific values that record the distribution measure in different degrees. In this section we only show the first two moments: the expected **hope** or value and the **variance** .

First moment: expected value.

Expected value of the Bernoulli distribution.

Second moment: variance.

Variance of the Bernoulli distribution.

## Example of Bernouilli’s moments

We assume that we want to calculate the first two moments of a Bernoulli distribution given a probability p = 0.6 such that

The frequency of the random variable D can approximate satisfactorily to a Bernoulli distribution.

Where D is a discrete random variable.

So, we know that p = 0.6 and that (1-p) = 0.4.

- First moment: expected value.

The expected value is the probability of success of the random variable D.

Second moment: variance.

Calculation of the variance of the Bernoulli distribution.

In addition, we want to calculate the distribution function given the probability p = 0.6. So:

We associate the start-up scheme with the results obtained.

Given the probability function:

Bernoulli distribution function.

### When z = 1

Bernoulli distribution function when z = 1.

### When z = 0

Bernoulli distribution function when z = 0.

The blue color indicates that the parts that coincide between both (equivalent) ways of expressing the probability distribution function of the Bernoulli distribution.