16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde If X and y were actually linearly If X and y were independent, the related, the points would appear on points would scatter all over the x,y one straight line, p would be tl, and plane, u would be zero, so the mean squared error in the =Y, and a2 approximation would be zero Note that dependence other than linear is not necessarily measured by p Example: y=X and X=X=0 u=Y=Y=X3-X2=0 >P=0, but X, Y are dependent Also, high correlation does not imply cause and effect Example: Dying in the hospital A survey reports that two events, "entering the hospital" and "dying within 1 week"have a high correlation. This relationship, however, is not causal. There exists a third, unreported event, " disease, which causes each of the other events Page 4 of 916.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde If X and Y were actually linearly If X and Y were independent, the related, the points would appear on points would scatter all over the x,y one straight line, ρ would be ±1 , and plane, µxy would be zero, so the mean squared error in the Y = Y , and ε 2 = σ y 2 . approx . approximation would be zero. Note that dependence other than linear is not necessarily measured by ρ. 2 3 Example: Y = X and X = X = 0 . 2 µxy = XY = XY = X 3 − X X = 0 → = ρ 0, but XY , are dependent! Also, high correlation does not imply cause and effect. Example: Dying in the hospital. A survey reports that two events, “entering the hospital” and “dying within 1 week” have a high correlation. This relationship, however, is not causal. There exists a third, unreported event, “disease,” which causes each of the other events. Page 4 of 9