Fundamentals of Probability and Statistics for Engineers

90 Fundamentals of Probability and Statistics for EngineersThis result leads immediately to an important generalization. Consider afunction of X and Y in the form g(X)h(Y ) for which an expectation exists.Then, if X and Y are independent,Efg…X†h…Y†g ˆEfg…X†gEfh…Y†g:…4:28†When the correlation coefficient of two random variables vanishes, we saythey are uncorrelated. It should be carefully pointed out that what we haveshown is that independence implies zero correlation. The converse, however, isnot true. This point is more fully discussed in what follows.The covariance or the correlation coefficient is of great importance in theanalysis of two random variables. It is a measure of their linear interdependencein the sense that its value is a measure of accuracy with which one randomvariable can be approximated by a linear function of the other. In order to seethis, let us consider the problem of approximating a random variable X by alinear function of a second random variable Y ,aY ‡ b, where a and b arechosen so that the mean-square error e, defined byis minimized. Upon taking partial derivatives of e with respect to a and b andsetting them to zero, straightforward calculations show that this minimum isattained whenande ˆ Ef‰X …aY ‡ b†Š 2 g; …4:29†a ˆ X Yb ˆ m X am YSubstituting these values into Equation (4.29) then gives 2 X 1 2 ) as theminimum mean-square error. We thus see that an exact fit in the mean-squaresense is achieved when jj ˆ1, and the linear approximation is the worst when ˆ 0. More specifically, when ˆ‡ 1, the random variables X and Y are saidto be positively perfectly correlated in the sense that the values they assume fallon a straight line with positive slope; they are negatively perfectly correlatedwhen ˆ 1and their values form a straight line with negative slope. Thesetwo extreme cases are illustrated in Figure 4.3. The value of jj decreases asscatter about these lines increases.Let us again stress the fact that the correlation coefficient measures only thelinear interdependence between two random variables. It is by no means ageneral measure of interdependence between X and Y. Thus, ˆ 0 does notimply independence of the random variables. In fact, Example 4.10 shows, theTLFeBOOK