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

Introduction; Review of Random Variables

Document Actions
  • Send this
  • Print this
  • 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} 3 \end{cases}\end{displaymath}Is this a random variable? \end{example}" align="bottom" border="0" height="303" width="554" />
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 November 23, 2009, 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
Reuse Course
Download this course