Difference between revisions of "ECE600 F13 Independent Random Variables mhossain" - Rhea

Latest revision as of 11:12, 21 May 2014

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.

P(\{X\in A\}\cap\{Y\in B\})=P(X\in A)P(Y\in B) \quad\forall A,B\in\mathcal F

There is an alternative definition of independence for random variables that is often used. We will show that X and Y are independent iff

f_{XY}(x,y)=f_X(x)f_Y(y)\quad\forall x,y\in\mathbb R

First assume that X and Y are independent and let A = (-∞,x], B = (-∞,y]. Then,

\begin{align} F_{XY}(x,y) &= P(X\leq x,Y\leq y) \\ &= P(X\in A,Y\in B) \\ &= P(X\in A)P(Y\in B) \\ &= P(X\leq x)P(Y\leq y) \\ &= F_X(x)F_Y(y) \\ \Rightarrow f_{XY}(x,y) &= f_X(x)f_Y(y) \end{align}

Now assume that f $_{XY}$ (x,y) = f $$ _X $$ (x)f $$ _Y $$ (y) ∀x,y ∈ R. Then, for any A,B ∈ B(R)

\begin{align} P(X\in A,Y\in B) &= \int_A\int_Bf_{XY}(x,y)dydx \\ &=\int_A\int_Bf_X(x)f_Y(y)dydx \\ &=\int_Af_X(x)dx\int_Bf_Y(y)dy \\ &= P(X\in A)P(Y\in B) \end{align}

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.

Questions and comments

If you have any questions, comments, etc. please post them on this page

Back to all ECE 600 notes

@@ Line 3: / Line 3: @@
 [[ECE600_F13_notes_mhossain|Back to all ECE 600 notes]]
+[[Category:ECE600]]
+[[Category:probability]]
+[[Category:lecture notes]]
+[[Category:slecture]]
 <center><font size= 4>
-'''Random Variables and Signals'''
+[[ECE600_F13_notes_mhossain|'''The Comer Lectures on Random Variables and Signals''']]
 </font size>
+[https://www.projectrhea.org/learning/slectures.php Slectures] by [[user:Mhossain | Maliha Hossain]]
 <font size= 3> Topic 12: Independent Random Variables</font size>
 </center>
 ----
+----
 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.
@@ Line 41: / Line 48: @@
 \end{align}</math></center>
-Thus, X and Y are inedependent iff f<math>_{XY}</math>(x,y) = f<math>_X</math>f)X<math>_Y</math>.
+Thus, X and Y are independent iff f<math>_{XY}</math>(x,y) = f<math>_X</math>(x)f<math>_Y</math>(y).

Difference between revisions of "ECE600 F13 Independent Random Variables mhossain" - Rhea

Latest revision as of 11:12, 21 May 2014

References

Questions and comments

Alumni Liaison