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\end{align}</math></center> | \end{align}</math></center> | ||
| − | Thus, X and Y are | + | Thus, X and Y are independent iff f<math>_{XY}</math>(x,y) = f<math>_X</math>f)X<math>_Y</math>. |
Revision as of 14:01, 12 November 2013
Random Variables and Signals
Topic 12: Independent Random Variables
We have previously defined statistical independence of two events A and b in F. We will now use that definition to define independence of random variables X and Y.
Definition $ \qquad $ Two random variables X and Y on (S,F,P) are statistically independent if the events {X ∈ A}, and {Y ∈ B} are independent ∀A,B ∈ F. i.e.
There is an alternative definition of independence for random variables that is often used. We will show that X and Y are independent iff
First assume that X and Y are independent and let A = (-∞,x], B = (-∞,y]. Then,
Now assume that f$ _{XY} $(x,y) = f$ _X $(x)f$ _Y $(y) ∀x,y ∈ R. Then, for any A,B ∈ B(R)
Thus, X and Y are independent iff f$ _{XY} $(x,y) = f$ _X $f)X$ _Y $.
References
- M. Comer. ECE 600. Class Lecture. Random Variables and Signals. Faculty of Electrical Engineering, Purdue University. Fall 2013.
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