# Any function is not a Random Variable

1. Feb 4, 2012

### zli034

There are plenty example of functions are random variables from my class note. I only interested of thinking up functions are not random variables.

This class is about set theory, probability measure, Borel sets, sigma algebra, and limsup, liminf of sets.

2. Feb 4, 2012

### mathman

Your question is confusing. Ordinary functions, such as polynomials, are not random variables, unless the terms themselves are.

3. Feb 4, 2012

### zli034

I guess the jump discontinuous functions are not random variables.

4. Feb 5, 2012

### Stephen Tashi

As mathman indicated, functions are not random variables. If you have a random variable you can define a "function of a random variable". It you have a random variable, then the random variable has an associated function called its "distribution" and it may have an associated function that is it's "density". Talking about a "function" not being a "random variable" is like talking about a "number" not being a "plane". They are completely different concepts.

Perhaps you are tyring to ask if there are any types of functions that are not distributions or densities for a random variable?

5. Feb 5, 2012

### lavinia

A random variable is just a measurable function on a measure space of total measure 1.

if the sigma algebra is the Lebesque measurable sets then any continuous function is measurable. E.g. any polynomial. But pretty much any function you can think of is Lebesque measurable. Non-measurable functions are difficult to come by.