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

Independence of r.v.s


\begin{definition}
$X$ and $Y$ are {\bf independent} if
\begin{displaymath}P(...
...) = P(X \in A) P(Y \in B) \forall A,B \in \Bc.
\end{displaymath}\end{definition}

  1. $X$ and $Y$ are independent iff $F_{XY}(a,b) = F_X(a)F_Y(b)
\forall (a,b) \in \Rbb^2$ .
  2. if $X$ and $Y$ are jointly continuous, then they are independent iff $f_{XY}(x,y) = f_X(x)f_Y(y)$
  3. If $X$ and $Y$ are discrete, then they are independent iff $p_{XY}(a,b) = p_X(a)p_Y(b)$ .
  4. If $X$ and $Y$ are jointly Gaussian random variables, then they are independent iff $\rho = 0$ . (Show this using the p.d.f.)
Caution: Gaussian r.v.s are special this way. As a general rule, uncorrelated does not imply independence. .

In practice, it is common to assume that random variables are independent based on physical arguments, rather than to prove is by identifying a joint density and computing the marginals.

Many times, independence is also taken as an assumption, even when it is not strictly true. This independence assumption frequently simplifies analysis. However, the validity of the assumption must be validated (e.g., using computer simulations).

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_4.html. This work is licensed under a Creative Commons License Creative Commons License