Personal tools
  •  
You are here: Home Electrical and Computer Engineering Stochastic Processes Introduction; Review of Random Variables

Introduction; Review of Random Variables

Document Actions
  • Content View
  • Bookmarks
  • CourseFeed

Probability   ::   Set Theory   ::   Definition   ::   Conditional   ::   Random   ::   Distribution   ::   R.V.S.

Random variables

Up to this point, the outcomes in $\Omega$ could be anything: they could be elephants, computers, or mitochondria, since $\Omega$ is simple expressed in terms of sets. But we frequently deal with numbers , and want to describe events associated with sets of numbers. This leads to the idea of a random variable.
\begin{definition}
Given a probability space $(\Omega,\Fc,P)$, a {\bf random va...
...\omega \in \Omega: X(\omega) \leq a\} \in \Fc.
\end{displaymath}\end{definition}
A function $X: \Omega \rightarrow \Rbb$ such that $\{w \in \Omega:
X(\omega) \leq a\} \in \Fc$ , that is, such that the events involved are in $\Fc$ , is said to be measurable with respect to $\Fc$ . That is, $\Fc$ is divided into sufficiently small pieces that the events in it can describe all of the sets associated with $X$ .


\begin{example}
Let $\Omega = \{1,2,3,4,5,6\}$, $\Fc =
\{\{1,2,3\},\{4,5,6\},\...
... \omega > 3
\end{cases}\end{displaymath}Is this a random variable?
\end{example}
We observe that a random variable cannot generate partitions of the underlying sample space which are not events in the $\sigma$ -field $\Fc$ .

Another way of saying that $X$ is measurable: For any $B \in \Bc$ ,

\begin{displaymath}\{\omega \in \Omega: X(\omega) \in B\} \in \Fc.
\end{displaymath}

Recalling the idea of Borel sets $\Bc$ associated with the real line, we see that a random variable is a measurable function from $(\Omega,\Fc)$ to $(\Rbb,\Bc)$ :

\begin{displaymath}X: (\Omega,\Fc) \rightarrow (\Rbb,\Bc).
\end{displaymath}

We will abbreviate the term random variable as r.v. .

We will use a notational shorthand for random variables.
\begin{definition}
Suppose $(\Omega,\Fc,P)$\ is a probability space and $X:(\Om...
...) = P(\{\omega \in \Omega: X(\omega) \in B\}).
\end{displaymath}\end{definition}
By this definition, we can identify a new probability space. Using $(\Rbb,\Bc)$ as the pre-probability space, we use the measure

\begin{displaymath}P_X(B) = P(X \in B).
\end{displaymath}

So we get the probability space $(\Rbb,\Bc,P_X)$ .

As a matter of practicality, if the sample space is $\Rbb$ , with the Borel field, most mappings to $\Rbb$ will be random variables.

To summarize:

\begin{displaymath}(\Omega,\Fc,P) \stackrel{X}{\longrightarrow } (\Rbb,\Bc,P_X)
\end{displaymath}

where

\begin{displaymath}P_X(B) = P(\{\omega \in \Omega \vert X(\omega) \in B\})
\end{displaymath}

for $B \in \Bc$ .

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 31). Introduction; Review of Random Variables. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lec1_5.html. This work is licensed under a Creative Commons License Creative Commons License