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# Uniform distribution

A continuous random variable has a uniform distribution if all the values belonging to its support have the same probability density.

## Definition

The uniform distribution is characterized as follows:

Definition Let be an absolutely continuous random variable. Let its support be a closed interval of real numbers:We say that has a uniform distribution on the interval if its probability density function is:

A random variable having a uniform distribution is also called a uniform random variable. Sometimes, we also say that it has a rectangular distribution or that it is a rectangular random variable.

To better understand the uniform distribution, you can have a look at its density plots.

## Expected value

The expected value of a uniform random variable is:

Proof

It can be derived as follows:

## Variance

The variance of a uniform random variable is:

Proof

We can use the variance formula as follows:

## Moment generating function

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

Proof

Using the definition of moment generating function:Note that the above derivation is valid only when . However, when :Furthermore, it is easy to verify that:When , the integral above is well-defined and finite for any . Thus, the moment generating function of a uniform random variable exists for any .

## Characteristic function

The characteristic function of a uniform random variable is:

Proof

Using the definition of characteristic function:Note that the above derivation is valid only when . However, when :Furthermore, it is easy to verify that:

## Distribution function

The distribution function of a uniform random variable is:

Proof

If , then:because can not take on values smaller than . If , then:If , then:because can not take on values greater than .

## Solved exercises

Below you can find some exercises with explained solutions:

1. Exercise set 1 (compute quantities related to the uniform distribution)

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