010+Correlation

For n random pairs xi yi the correlation coecient is

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Unformatted text preview: E (y ))] V ar(x) · V ar(y ) , where V ar(x) = E (x − E (x))2 , V ar(y ) = E (y − E (y ))2 . ρ is the theoretical moment related to probability density function f (x, y ) or P (X = x, Y = y ). For n random pairs (xi , yi ) the correlation coecient is r= SSxy , S Sxx SSyy 3 A. Zhensykbaev Correlation where n n SSxy = (xi − x)(yi − y ) = ¯ ¯ i=1 i=1 n n 2 SSxx = x2 i (xi − x) = ¯ i=1 n i=1 n 2 SSyy = 2 yi (yi − y ) = ¯ i=1 x= ¯ i=1 1 n n xi , y= ¯ i=1 1 − n 1 − n 1 n n n 1 xi y i − n xi yi , i=1 n i=1 2 xi i=1 n , 2 yi , i=1 n xi . i=1 Main properties: 1. −1 ≤ ρ ≤ 1 2. if X and Y are independent, then ρ = 0; 3. if ρ = 0, X and Y are normally distributed then they are independent; 4. if ρ = ±1 then X and Y are linearly dependent (Y = kX , k is constant). Remark. In general case property 3. is not true: ρ = 0, then X and Y can be both dependent and independent. For example, Consider two independent random variables X and Y such that E (X ) = E (Y ) = 0, and variable Z = X · Y . It is clear that Z is dependent on X . Find the correlation coecient ρ: ρ...
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This note was uploaded on 02/11/2014 for the course MATH 1390 taught by Professor Christopherstocker during the Spring '13 term at Marquette.

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