Random Variables and Signals
Topic 5: Random Variables: Definition
Random Variables
So far, we have been dealing with sets. But as engineers, we work with numbers, variables, vectors, functions, signals, etc. We will spend the rest of the course learning about random variables, vectors, sequences, and processes, since these usually serve us better as engineers in practice. However, we will link each of these topics to a probability space, since probability spaces tie everything together.
Informally, a random variable, often abbreviated to rv, is a function mapping the sample space S to the reals. However, we will restrict this function to lie within a certain class of functions.
First we present some definitions that will be needed.
Definition $ \quad $ Given two spaces S and R, an R-valued function or mapping $ f $:S→R assigns each element in S a value in R, so
Definition $ \quad $ Given any F ⊂ S and G ⊂ R, the image of F under $ f $ is
and the inverse image of G under $ f $ is
Note the following
- $ f $(F) is the set of all values in R that are mapped to by some element in F
- $ f^{-1} $(G) is the set of all values in S that map to some value in G
- The inverse image, $ f^{-1} $(G), is not the same as the inverse function $ f^{-1} $(a), a ∈ R.
- $ f(\omega) $, $ \omega $ ∈ S, is an element of R; $ f $(F), F ⊂ S, is a subset of R.
- $ f^{-1} $(a), a ∈ R, is an element of S; $ f^{-1} $(G), G ⊂ R, is a sunset of S.
- The inverse function $ f^{-1} $(a) ∀a ∈ R, may not exist. The inverse image $ f^{-1} $(G), G ⊂ R, always exists.
Definition of Random Variable (Beta version)</br>
Given a probability space (S,F.p), a random variable is a function from S to the real line,
References
- M. Comer. ECE 600. Class Lecture. Random Variables and Signals. Faculty of Electrical Engineering, Purdue University. Fall 2013.
