131A_1_Lecture11-2_Winter_2012

131A_1_Lecture11-2_Winter_2012 - EE 131A Probability...

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UCLA EE131A (KY) 1 EE 131A Probability Professor Kung Yao Electrical Engineering Department University of California, Los Angeles Lecture 11-2

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UCLA EE131A (KY) 2 Independence of two rv’s (1) Review: Two events A and B are independent if P(A B) = P(A)P(B) . Now, we want to consider the independence concept from events to random variables. The two rv’s X and Y are independent if the joint cdf satisfies F XY (x,y) = F X (x) F Y (y), - < x < , - < y < , or equivalently the joint pdf satisfies f XY (x,y) = f X (x) f Y (y), - < x < , - < y < .
UCLA EE131A (KY) 3 Independence of two rv’s (2) Ex. 1. (Ex. 2, p. 10, Lec10-2). Then Thus, X and Y are two independent rv’s. Ex. 2. (Ex. 2, p. 4, Lec11-1). f XY (x,y) f X (x) f Y (y) , - < x < , - < y < . Thus, X and Y are not independent rv’s. XY XY X Y XY X Y f (x) = 1, 0 x < 1, f (y) = 1, 0 y < 1. f (x,y) = f (x) f (y), 0 x < 1, 0 y < 1, F (x,y) = F (x) F (y), 0 x < 1, 0 y < 1  XY 1, 0 x 1, 0 y 1, f( x , y ) = 0, elsewhere .

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UCLA EE131A (KY) 4 Independence of two rv’s (4) Ex. 3. (Ex. 3, p. 6, Lec11-1). For this problem, recall Y \ X 1234 0 (B) .12
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131A_1_Lecture11-2_Winter_2012 - EE 131A Probability...

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