Independent uncorrelated independent uncorrelated y

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Unformatted text preview: ndent vs. Uncorrelated Sum of Independent Random Vars. • Independent • Uncorrelated • Independent Uncorrelated. y • Uncorrelated Independent?? – Not true, in general. – True for multivariate Gaussian. z+dz dy 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.

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