251v2out 4/19/2006 (Open this document in 'Outline' view!)K. Two Random Variables.1. Regression (Summary).2. Covariance ( and )a. Population Covarianceb. The Sample Covariance3. The Correlation Coefficient ( and )a. Population Correlation.b. Sample Correlation.4. Functions of Two Random Variables.5. Sums of Random Variables.a. andb. Independence.(i) Definition.(ii) Consequencesd. Application to portfolio analysis251v2out 4/19/2006 (Open this document in 'Outline' view!)K. Two Random Variables.1. Regression (Summary).2. Covariance (xy and sxy)a. Population CovarianceThe population covariance is defined, using probability, as yxyxxyxyEyxEyxCov),(. This can be used to describe the relationship between x and y. If the covariance is positive we can say that x and y tend to move together, while if it is negative we can say that they tend to move in opposite directions. In order to use this formula we must realize that yxxyPxyE ,. This means that we must add together the product of x and y, together with their joint probability, for each possible pair of values of x andy. For example, assume that x and y are related by the following joint probability table:09.07.16.08.05.10.18.15.12.800600400800600400yx.We begin by taking the upper left hand probability, .12, which is the probability that both x and y are 400, and multiplying it by 400 twice. Then we take the next probability in the same row, .15, which is the probability that x is 600 and y is 400, and multiply it by both 600 and 400. If we continue in this way we get .33560057600336005120038400180002400057600360001920080080009.80060007.80040016.60080008.60060005.60040010.40080018.40060015.40040012.xyxyPxyEWe can now use the following tableau to compute the means and variances of x and y. 35960057438200059400.1224000972006080028016215235.27.38.204800828007200025613818032.23.45.09.07.16.08.05.10.18.15.12.80060040080060040022xPxxxPxPyyPyyy PyPx2To summarize 1xP (a check), 594xxPxEx, 38200022xPxxE, 1yP, 574yyPyEy and 35960022yPyyEWe will need the variances below. To complete what we have done, write 5356574594335600 yxxyxyExyCovb. The Sample CovarianceThe sample covariance is much easier to compute, the formula being 11 nyxnxynyyxxsxy.For example, assume that we have data on income (x) and savings (y)(in thousands) for 5 families.Family xy 2x2yxy1 1.9 0.0 3.61 0.00 0.002 12.4 0.9 153.76 0.81 11.163 6.4 0.4 40.96 0.16 2.564 7.0 1.2 49.00 1.44 8.405 7.0 0.3 49.00 0.09 2.10Then 94.657.34xand 56.058.2y. 878.13494.6533.29612222nxnxsx, 2330.0456.0550.212222nynysy and since 22.24xy, 197.11556.094.6522.24xys. The positive sign of xys, the sample covariance, indicates that x and y tend to move together.3Sum 34.7 2.8 296.33 2.50 24.223. The Correlation Coefficient ( xy and xyr)The size of a covariance is relatively meaningless; to judge the strength of the relationship between x and y we need to compute the correlation, which is found by dividing the covariance by the standard deviations of x and y. a. Population Correlation. For the population covariance, recall from above that 291645943820002222xxxE and 301245743596002222yyyE. So that 181.056.17377.170535630124291645356yxxyxy. The correlation must always be between positive and negative 1 0.10.1 . A correlation close to zero is called weak. A correlation that is close to one in absolute value is called strong. (Actually statisticians prefer to look at the value of the correlation squared.) A strong positive correlation indicates that x and y have a relationship that is close to a straight line with a positive slope. A strong negative correlation means that the relationship approximates a straight line with a negative slope. Unfortunately, the correlation only indicates linear relationships; a nonlinear relationship that is obvious on a graph may give a zero correlation.b. Sample Correlation.Recall that 2330.0 and ,878.13,197.122yxxysss. If we divide thecorrelation by the two standard deviations, we find that .6657.02330.0878.13197.1yxxyxysssr4. Functions of Two Random Variables.),(),( yxacCovdcybaxCov and if dcybaxw vand , xywvacsign or),())((),( yxCorracsigndcybaxCorr , where acsign has the value 1 or 1depending on whether the product of a and c is negative or positive.5. Sums of Random Variables.a. yExEyxE and xyyxyxVar222 yxCovyVarxVar ,2b. Independence.(i) Definition. 4 yPxPyxP ,(ii) Consequences If x y and are independent, yExExyE , 0, yxCov, 0xyand yVarxVaryxVar .c. If ,ac and d are constants, ),(2)()()(22yxacCovyVarcxVaracyaxVar . This and a. imply that dycExaEdcyaxE and ),(2)()()(22yxacCovyVarcxVaradcyaxVar d. Application to portfolio analysisIf 1 and 212211 PPRPRPR, then 2211REPREPRE and 2121222121,2 RRCovPPRVarPRVarPRVar . Variance is usually considered a measureof risk, though actually, the best measure of risk is probably the coefficient of variation, the standard deviation divided by the mean, in this case RECR.The remainder of this material can be found in the Supplement in the document 251var2. 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