Estimate the coefficient of correlation for the data of Example 4.
From the PDFs given in Example 11 it can be easily shown that ax = 2.05 and ay = 1.50. We have already shown that cov(X, Y) = 2.24. Therefore, applying the preceding formula we estimate p as 2.24/(2.05)(1.50) = 0.73.
Variances of Correlated Variables. Let X and Y be two rv's. Then var (X + Y) = var (X) + var (Y) + 2 cov (X, Y)
= var (X) + var (Y) + 2poxoy var (X — Y) = var (X) + var (Y) — 2 cov (X, Y) = var (X) + var (Y) — 2poxoy
If, however, X and Y are independent, cov (X, Y) is zero, in which case the var(X + Y) and var(X — Y) are both equal to var(X) + var(Y), as noted previously.
The preceding results can be generalized as follows. Let Ya=\Xi = X1 + X2 +----+ Xn, then the variance of the linear combination J^Xi is
= y] var Xi + pijOiOj i=1 i<j where pij is the correlation coefficient between Xi and Xj and where Oi and Oj are the standard deviations of Xi and Xj .
884 APPENDIX A: A REVIEW OF SOME STATISTICAL CONCEPTS
Thus, var (X1 + X2 + X3) = varX + varX2 + var X3 + 2 cov(X1, X2) + 2 cov(X1, X3) + 2 cov(X2, X3) = var X1 + var X2 + var X3 + 2p12a1a2 + 2poaa + 2p23a2a3
where a1, a2, and a3 are, respectively, the standard deviations of X1, X2, and X3 and where p12 is the correlation coefficient between X1 and X2, p13 that between X1 and X3, and p23 that between X2 and X3.
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