131A_1_Lecture13-1_Winter_2012

131A_1_Lecture13-1_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 13-1
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UCLA EE131A (KY) 2 Distribution and pdf of (X+Y) (1) When we have two rv’s, X and Y, often they may interact in the form of (X + Y). Suppose we have the joint pdf f XY (x,y) of these rv’s. Denote Z = X + Y. We want to find the cdf F Z (z) and the pdf f Z (z) of Z. First, start with cdf F Z (z). We note, the above three integrals are not easy. But, how about the pdf f Z (z) ? Z z-y z-x XY XY XY x+y z x=- y=- y=- x=- F (z) = P(Z z) = P(X+Y z) = P(Y z-X) = P(X z-Y) = f (x,y)dxdy = f (x,y)dy dx = f (x,y)dx dy       
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UCLA EE131A (KY) 3 Distribution and pdf of (X+Y) (2) The above two integrals are not easy either. Now, suppose X and Y are two independent rv’s. Use (2) in (1) The integrals of (3) and (4) are both convolutions!
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This note was uploaded on 02/29/2012 for the course ELECTRICAL 131a taught by Professor Yao during the Spring '12 term at UCLA.

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131A_1_Lecture13-1_Winter_2012 - EE 131A Probability...

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