A continuous random variable has a uniform distribution if all the values belonging to its support have the same probability density.
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.
The expected value of a uniform random variable is:
It can be derived as follows:
The variance of a uniform random variable is:
We can use the variance formula as follows:
The moment generating function of a uniform random variable is defined for any :
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 .
The characteristic function of a uniform random variable is:
Using the definition of characteristic function:Note that the above derivation is valid only when . However, when :Furthermore, it is easy to verify that:
The distribution function of a uniform random variable is:
If , then:because can not take on values smaller than . If , then:If , then:because can not take on values greater than .
Below you can find some exercises with explained solutions:
Exercise set 1 (compute quantities related to the uniform distribution)
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