16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Lecture 5 Last time Characterizing groups of random variables Names for groups of random variables S=∑X s2=∑∑xX Characterize by pairs to compute E[x]=X=「d观(xy which we define as the correlation Often we do not know the complete distribution, but only simple statistics The most common of the moments of higher ordered distribution functions is the Covariance, Hn=E(X-X)(Y-)=(x-X)( XY-Xy+ Xy XY-X (correlation)-(product of means) Even more significant is the normalized covariance, or correlation coefficient: 1≤p≤1 o201o This correlation coefficient may be thought of as measuring the degree of linear dependence between the random variables: P=0 if the two are independent and p=+l if one is a linear function of the other. First note p=0 if X and Y are ndependent Calculate p for Y=a+bX If linearly related16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Lecture 5 Last time: Characterizing groups of random variables Names for groups of random variables n S = ∑Xi i=1 n n 2 S = ∑∑X iX j i=1 j=1 Characterize by pairs to compute ∞ ∞ E XY ] = XY = dx xyf xy [ ( , ) x y dy ∫ ∫ , −∞ −∞ which we define as the correlation. Often we do not know the complete distribution, but only simple statistics. The most common of the moments of higher ordered distribution functions is the covariance, ⎣ )( ⎦ µ )( xy = E ⎡( X − XY −Y )⎤ = ( X − XY −Y ) = XY − XY − XY + XY = XY − XY − XY + XY = XY − XY = (correlation) − (product of means) Even more significant is the normalized covariance, or correlation coefficient: µxy ρ = µ 2 xy 2 = σ σ , −1 ≤ ρ ≤ 1 σ σ x y x y This correlation coefficient may be thought of as measuring the degree of linear dependence between the random variables: ρ = 0 if the two are independent and ρ = ±1 if one is a linear function of the other. First note ρ = 0 if X and Y are independent. Calculate ρ for Y a = + bX . xy If linearly related: Page 1 of 9