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WCU ECO 251 - Two Random Variables

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251v2outl 4 19 2006 Open this document in Outline view K Two Random Variables 1 Regression Summary 2 Covariance xy and s xy a Population Covariance The population covariance is defined using probability as Cov x y xy E x x y y E xy x y 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 E xy xyP x y 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 and y For example assume that x and y are related by the following joint probability table y 400 600 800 400 12 10 16 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 E xy xyP xy 12 400 400 15 600 400 18 800 400 10 400 600 05 600 600 08 800 600 16 400 800 07 600 800 09 800 800 36000 57600 19200 24000 18000 38400 335600 51200 33600 57600 We can now use the following tableau to compute the means and variances of x and y x 600 15 05 07 800 18 08 09 400 y 600 800 P x xP x x 2 P x x 400 600 800 12 15 18 10 05 08 16 07 09 38 27 35 152 162 280 60800 97200 224000 P y 45 23 32 yP y 180 138 256 y P y 72000 82800 204800 2 1 00 574 359600 594 382000 2 To summarize P x 1 a check xP x 594 E x x P x 382000 P y 1 E y yP y 574 and E y y P y 359600 x E x 2 2 y 2 2 We will need the variances below To complete what we have done write xy Cov xy E xy x y 335600 594 574 5356 b The Sample Covariance The sample covariance is much easier to compute the formula being s xy x x y y n 1 xy nx y n 1 For example assume that we have data on income savings y in thousands for 5 families Family 1 2 3 4 5 Sum x y 1 9 12 4 6 4 7 0 7 0 34 7 0 0 0 9 0 4 1 2 0 3 2 8 x 34 7 y 2 8 x and xy 24 22 Then x 2 x and x2 3 61 153 76 40 96 49 00 49 00 296 33 296 33 y2 0 00 0 81 0 16 1 44 0 09 2 50 y 2 xy 0 00 11 16 2 56 8 40 2 10 24 22 2 50 34 7 2 8 6 94 and y 0 56 5 5 x 2 s x2 y 2 s 2y nx 2 n 1 ny 2 n 1 xy 24 22 296 33 5 6 94 2 13 878 4 2 50 5 0 56 2 0 2330 and since 4 s xy 24 22 5 6 94 0 56 1 197 5 1 The positive sign of s xy the sample covariance indicates that x and y tend to move together 3 3 The Correlation Coefficient xy and rxy 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 x2 E x 2 x2 382000 594 2 29164 and 2y E y 2 y2 359600 574 2 30124 So that xy 5356 0 181 x y 29164 30124 170 77 173 56 The correlation must always be between positive 1 and negative 1 1 0 1 0 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 xy 5356 b Sample Correlation 2 2 Recall that s xy 1 197 s x 13 878 and s y 0 2330 If we divide the correlation by the two standard deviations we find that rxy s xy sx s y 1 197 13 878 0 2330 0 6657 4 Functions of Two Random Variables Cov ax b cy d acCov x y and if w ax b and v cy d wv sign ac xy or Corr ax b cy d sign ac Corr x y where sign ac has the value 1 or 1 depending on whether the product of a and c is negative or positive 5 Sums of Random Variables a E x y E x E y and 4 Var x y x2 y2 2 xy Var x Var y 2Cov x y b Independence i Definition P x y P x P y If ii Consequences x and y are independent E xy E x E y Cov x y 0 xy 0 and Var x y Var x Var y c If a c and d are constants Var ax cy a 2Var x c 2Var y 2acCov x y This and a imply that E ax cy d aE x cE y d and Var ax cy d a 2Var x c 2Var y 2acCov x y d Application to portfolio analysis Most of this is from the document 251var2 in the supplement If R P1 R1 P2 R2 and P1 P2 1 then E R P1 E R1 P2 E R 2 and Var R P12Var R1 P22Var R2 2 P1 P2 Cov R1 R2 is the variance of the return Thus if P1 and P2 are both 50 we can say Var R 25Var R1 25Var R 2 50Cov R1 R 2 For example assume that R1 0 20 R2 0 30 but R1R 2 is unknown Then Cov R1 R 2 R1R2 R1 R2 R1R2 20 30 06 R1R2 If we use the formula for Var R immediately above Var R 25 20 2 25 30 2 50 06 R1R2 0100 0225 0300 R1R 2 0325 0300 R1R 2 Now we can see the effect various values of …


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