Pattern Recognition and Machine Learning

Exercises 603

12.29 (**)EmSupposethattwovariablesZlandZ2areindependentsothatp(zl'Z2)=
P(Zl)P(Z2)' Showthatthecovariancematrixbetweenthesevariablesisdiagonal.
Thisshowsthatindependenceisa sufficientconditionfortwovariablestobeun-

correlated. NowconsidertwovariablesYlandY2 inwhich-1 :0;; Yl :0;; 1 and

Y2 =yg. Writedowntheconditionaldistributionp(Y2IYl)andobservethatthisis

dependentonYbshowingthatthetwovariablesarenotindependent. Nowshow

thatthecovariancematrixbetweenthesetwovariablesisagaindiagonal.Todothis,

usetherelationP(Yl,Y2) =P(YI)p(Y2IYl)toshowthattheoff-diagonaltermsare

zero.Thiscounter-exampleshowsthatzerocorrelationisnota sufficientcondition

forindependence.