The support of a random variable is the set of values that the random variable can take.

For discrete random variables, it is the set of all the realizations that have a strictly positive probability of being observed.

Example If a discrete random variable has probability mass functionits support, denoted by , is

For continuous random variables, it is the set of all numbers whose probability density is strictly positive.

Example If a continuous random variable has probability density functionthen its support is

The same definition applies to random vectors. If is a random vector, its support is the set of values that it can take. The concept extends in the obvious manner also to random matrices.

The support is sometimes also called **range**.

The lecture entitled Random variables explains the concept of support in more detail.

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Please cite as:

Taboga, Marco (2021). "Support of a random variable", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/support-of-a-random-variable.

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