# 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 beLet . 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, we getObviously, 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, we obtain

## 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.

### Exercise 1

Let and be two independent Bernoulli random variables with parameter . Derive the probability mass function of their sum

Solution

The probability mass function of isThe probability mass function of isThe support of (the set of values can take) isThe convolution formula for the probability mass function of a sum of two independent variables iswhere is the support of . When , the formula givesWhen , the formula givesWhen , the formula givesTherefore, the probability mass function of is

### Exercise 2

Let be a Bernoulli random variable with parameter . Find its tenth moment.

Solution

The moment generating function of isThe tenth moment of is equal to the tenth derivative of its moment generating function, evaluated at :Butso that

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