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131A_1_Lecture14-1_Winter_2012

# 131A_1_Lecture14-1_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 14-1

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UCLA EE131A (KY) 2 Higher-dimensional Gaussian pdf’s Since Gaussian rv is one of the most important rv’s in probability, applied science, and engineering, we want to study the higher dim. Gaussian random vectors. 1. One-dimensional case The mean is a translation parameter and the variance 2 is a width scaling parameter. 2 2 2 -(x- μ ) /(2 σ ) X f (x) = (1/ 2 πσ )e , <x< .  -4 -2 0 2 4 6 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8     x
UCLA EE131A (KY) 3 Two-dimensional Gaussian pdf 2. Two dimensional case with X 1 and X 2 . 1 is the mean of X 1 , 2 is the mean of X 2 . 2 1 is the variance of X 1 and 2 2 is the variance of X 2 . is the correlation coefficient between X 1 and X 2 . 2 2 1 1 1 1 2 2 2 2 2 1 1 2 2 1 2 -1 x - μ x - μ x - μ x - μ 2 ρ σ σ σ σ 2(1 ρ ) 1 X X 1 2 2 2 1 2 - < x < , e , f (x ,x ) = - < x < . 2 πσ σ 1 ρ    

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