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More on Random Variables

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Expectation  ::  Properties  ::  Pairs  ::  Independence  ::  Two R.V.S.  ::  Functions  ::  Inequalities  ::  Conditional  ::  General

Properties of Expectations

  1. If $X=c$ then $E[X] = c$ .
  2. If $Y = aX+b$ then $E[Y] = aE[X] + b$ .

$E[X]$ acts kind of like an integral of $X(\omega)$ over $\Omega$ , weighted by $P$ . One way that the expectation is expressed is

\begin{displaymath}E[X] = \int_\Omega X(\omega) P(d\omega) = \int_\Omega X dP.
\end{displaymath}

An integral in this form is said to be a Lebesgue-Stieltjes Integral. Since $X$ induces a probability $P_X$ on $(\Rbb,\Bc)$ , as we have observed we can also think of the probability space $(\Rbb,\Bc,P_X)$ . We can write

\begin{displaymath}E[X] = \int_\Rbb x P_X(dx)
\end{displaymath}

where now $X$ is the ''identity'' r.v. on the real line. We thus have two equivalent definitions:

\begin{displaymath}\int_\Omega X(\omega) P(d\omega) = \int_\Rbb x P_X(dx)
\end{displaymath}

Back to properties:

  1. If $Y = g\circ X$ then

    \begin{displaymath}E[Y] = \int_{\Omega} (g \circ X)(\omega) P(d\omega) = \int_\Rbb
g(x) P_X(dx) = \int_\Rbb y P_Y(dy)
\end{displaymath}

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, May 31). More on 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/lec2_2.html. This work is licensed under a Creative Commons License Creative Commons License