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# Bernoulli distribution

Suppose you perform an experiment with two possible outcomes: either success or failure. Success happens with probability , while failure happens with probability . A random variable that takes value in case of success and in case of failure is called a Bernoulli random variable (alternatively, it is said to have a Bernoulli distribution).

## Definition

Bernoulli random variables are characterized as follows:

Definition Let be a discrete random variable. Let its support be:Let . We say that has a Bernoulli distribution with parameter if its probability mass function is:

A random variable having a Bernoulli distribution is also called a Bernoulli random variable.

Note that, by the above definition, any indicator function is a Bernoulli random variable.

The following is a proof that is a legitimate probability mass function:

Proof

Non-negativity is obvious. We need to prove that the sum of over its support equals . This is proved as follows:

## Expected value

The expected value of a Bernoulli random variable is:

Proof

It can be derived as follows:

## Variance

The variance of a Bernoulli random variable is:

Proof

It can be derived thanks to the usual variance formula ():

## Moment generating function

The moment generating function of a Bernoulli random variable is defined for any :

Proof

Using the definition of moment generating function:Obviously, the above expected value exists for any .

## Characteristic function

The characteristic function of a Bernoulli random variable is:

Proof

Using the definition of characteristic function:

## Distribution function

The distribution function of a Bernoulli random variable is:

Proof

Remember the definition of distribution function:and the fact that can take either value or value . If , then , because can not take values strictly smaller than . If , then , because is the only value strictly smaller than that can take. Finally, if , then , because all values can take are smaller than or equal to .

## More details

### Relation between the Bernoulli and the binomial distribution

A sum of independent Bernoulli random variables is a binomial random variable. This is discussed and proved in the lecture entitled Binomial distribution.

## Solved exercises

Below you can find some exercises with explained solutions:

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