Stat 5101 Lecture Notes - School of Statistics

52 Stat 5101 (Geyer) Course NotesIf var(X) orvar(Y) is zero, the correlation is undefined.Again we might ask why two such closely related concepts as correlation andcovariance. Won’t just one do? (Recall that we asked the same question aboutvariance and standard deviation.) Here too we have the same answer. Thecovariance is simpler to handle theoretically. The correlation is easier to understandand hence more useful in applications. Correlation has three importantproperties.First, it is a dimensionless quantity, a pure number. We don’t think muchabout units, but if we do, as we noted before the units X and sd(X) are thesame and a little thought shows that the units of cov(X, Y ) are the product ofthe units of X and Y . Thus in the formula for the correlation all units cancel.Second, correlation is unaltered by changes of units of measurement, that is,cor(a + bX, c + dY ) = sign(bd) cor(X, Y ), (2.33)where sign(bd) denotes the sign (plus or minus) of bd. The proof is left as anexercise (Problem 2-25).Third, we have the correlation inequality.Theorem 2.24 (Correlation Inequality). For any random variables X andY for which correlation is defined−1 ≤ cor(X, Y ) ≤ 1. (2.34)Proof. This is an immediate consequence of Cauchy-Schwarz. Plug in X − µ Xfor X and Y − µ Y for Y in (2.32), which is implied by Cauchy-Schwarz by thecomment following the proof of the inequality, giving|cov(X, Y )| ≤ √ var(X) var(Y ).Dividing through by the right hand side gives the correlation inequality.The correlation has a widely used Greek letter symbol ρ (lower case rho). Asusual, if correlations of several pairs of random variables are under consideration,we distinguish them by decorating the ρ with subscripts indicating the randomvariables, for example, ρ X,Y = cor(X, Y ). Note that by definition of correlationcov(X, Y ) = cor(X, Y )sd(X)sd(Y)=ρ X,Y σ X σ YThis is perhaps one reason why covariance doesn’t have a widely used Greeklettersymbol (recall that we said the symbol σ X,Y used by Lindgren is nonstandardand not understood by anyone who has not had a course using Lindgrenas the textbook).Problems2-1. Fill in the details at the end of the proof of Corollary 2.2. Specifically,answer the following questions.