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

Latest revision as of 12:12, 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.

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Back to all ECE 600 notes

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 [[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.
+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''' <math>\qquad</math> 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. <br/>
@@ 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 12:12, 21 May 2014

References

Questions and comments

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