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Unformatted text preview: ndent vs. Uncorrelated Sum of Independent Random Vars. • Independent
• Independent Uncorrelated.
y • Uncorrelated Independent??
– Not true, in general.
– True for multivariate Gaussian. z+dz
dx x Y Y Y z =x + y
43 44 Correlation Coefficient Central Limit Theorem
Given n independent random variables X i , i 1, , n, EX i mi ,VarX i i2 ,
n n i 1 i 1 Z X i , EZ i mz mi • Normalized measure of correlation between . • Value between 1 and 1 n VarZ i z2 i2 , • Zero for uncorrelated i 1 Property of Convolutions: Convolution of a large number
of positive functions is approximately Gaussian.
Central Limit Theorem: Z is asymptotically Gaussian. _|Å F ( z) N m , Z z 2
z n 45 Zero Correlation Coefficient • Reduces to variance property for 46 Unity Correlation Coefficient • Uncorrelated • For uncorrelated
47 48 Range of Correlation Coefficient Orthogonal Random Variables
∗ Proof: Quadratic in a has no real roots negative or zero discriminant for quadratic in a
(zero for X = Y, equal roots) 49 Correlation 50 Covariance Matrix and Covariance Generalization of 2nd moment &...
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This document was uploaded on 02/23/2014 for the course EE 782 at University of Nevada, Reno.
- Fall '13
- Electrical Engineering