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Index > Probability distributions

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 X be an absolutely continuous random variable. Let its support be a closed interval of real numbers:[eq1]We say that X has a uniform distribution on the interval $left[ l,u
ight] $ if its probability density function is:[eq2]

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 X is:[eq3]

Proof

It can be derived as follows:[eq4]

Variance

The variance of a uniform random variable X is:[eq5]

Proof

We can use the variance formula [eq6] as follows:[eq7]

Moment generating function

The moment generating function of a uniform random variable X is defined for any t in R:[eq8]

Proof

Using the definition of moment generating function:[eq9]Note that the above derivation is valid only when $t
eq 0$. However, when $t=0$:[eq10]Furthermore, it is easy to verify that:[eq11]When $t
eq 0$, the integral above is well-defined and finite for any t in R. Thus, the moment generating function of a uniform random variable exists for any t in R.

Characteristic function

The characteristic function of a uniform random variable X is:[eq12]

Proof

Using the definition of characteristic function:[eq13]Note that the above derivation is valid only when $t
eq 0$. However, when $t=0$:[eq14]Furthermore, it is easy to verify that:[eq15]

Distribution function

The distribution function of a uniform random variable X is:[eq16]

Proof

If $x<l$, then:[eq17]because X can not take on values smaller than $l$. If $lleq xleq u$, then:[eq18]If $x>u$, then:[eq19]because X can not take on values greater than $u$.

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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