Correlation in Random Variables

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Correlation in Random VariablesLecture 11Spring 2002Correlation in Random VariablesSuppose that an experimentproduces two Random vari-ables, about the relationship be-tween them?One of the best ways to visu-alize the possible relationshipis to plot the (X, Y)pairthatis produced by several trials ofthe experiment. An exampleof correlated samples is shownat the rightLecture 111Joint Density FunctionThe joint behavior ofXandYis fully captured in the joint probabilitydistribution. For a continuous distributionE[XmYn]= xmynfXY(x, y)dxdyFor discrete distributionsE[XmYn]= x Sx y SyxmynP(x, y)Lecture 112Covariance FunctionThe covariance function is a number that measures the commonvariation is defined ascov(X, Y)=E[(X E[X])(Y E[Y])]=E[XY] E[X]E[Y]The covariance is determined by the difference inE[XY]andE[X]E[Y].IfXandYwere statistically independent thenE[XY] would equalE[X]E[Y] and the covariance would be covariance of a Random variable with itself is equal to its [X, X]=E[(X E[X])2]=var[X]Lecture 113Correlation CoefficientThe covariance can be normalized to produce what is known as thecorrelation coefficient.

Correlation Coefficient The covariance can be normalized to produce what is known as the correlation coefficient, ρ. ρ = cov(X,Y) var(X)var(Y) The correlation coefficient is bounded by −1 ≤ ρ ≤ 1. It will have value ρ = 0 when the covariance is zero and value ρ = ±1 when X and Y are perfectly correlated or anti-correlated. Lecture 11 4

  Correlations, Covariance