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Introduction; Review of Random Variables

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

Set theory

Probability is intrinsically tied to set theory. We will review some set theory concepts.

We will use $^c$ to denote complementation of a set with respect to its universe.

$\cup$ - union: $A \cup B$ is the set of elements that are in $A$ or $B$ .

$\cap$ - intersection: $A \cap B$ is the set of elements that are in $A$ and $B$ . We will also denote this as $AB$ .

$a \in A$ : $a$ is an element of the set $A$ .

$A \subset B$ : $A$ is a subset of $B$ .

$A=B$ : $A \subset B$ and $B \subset A$ .

Note that

\begin{displaymath}A \cup A^c = \Omega
\end{displaymath}

(where $\Omega$ is the universe).

Notation for some special sets:

$\Rbb$ - set of all real numbers

$\Zbb$ - set of all integers

$\Zbb^+$ - set of all positive integers

$\Nbb$ - set of all natural numbers, 0,1,2,...,

$\Rbb^n$ - set of all $n$ tuples of real numbers

$\Cbb$ - set of complex numbers


\begin{definition}
A {\bf field} (or algebra) of sets is a collection of sets that is
closed under complementation and finite union.
\end{definition}
That is, if $\Fc$ is a field and $A \in \Fc$ , then $A^c$ must also be in $\Fc$ (closed under complementation). If $A$ and $B$ are in $\Fc$ (which we will write as $A,B \in \Fc$ ) then $A \cup B \in \Fc$ .

Note: the properties of a field imply that $\Fc$ is also closed under finite intersection. (DeMorgan's law: $AB = (A^c \cup B^c)^c$ )


\begin{definition}
A $\sigma$-field (or $\sigma$-algebra) of sets is a field th...
...
also closed under {\em countable} unions (and intersections).
\end{definition}
What do we mean by countable?

  • A set with a finite number in it is countable.
  • A set whose elements can be matched one-for-one with $\Zbb$ is countable (even if it has an infinite number of elements!)
Are there non-countable sets?

Note: For any collection $F$ of sets, there is a $\sigma$ -field containing $F$ , denoted by $\sigma(F)$ . This is called the $\sigma$ -field generated by $F$ .

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_2.html. This work is licensed under a Creative Commons License Creative Commons License