**The Comer Lectures on 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 $(x)f$ _Y $(y).

## References

- M. Comer. ECE 600. Class Lecture. Random Variables and Signals. Faculty of Electrical Engineering, Purdue University. Fall 2013.

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