KIC000015 - Example l> Example Continued Suppose we have...

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Example l> Suppose we have two random variables, X and Y. X has a mean of 0 and a variance of I. Y has a mean of2 and a variance of 4. The covariance between X and Yis-1. ). Define the random variable Z as Z=2X+3Y ). Find the mean and variance of Z. Estimators of Mean, Variance, and Covariance ). Ifwe have a sample of N data points in hand, then we can compute the sample mean X using the following formula: I ,. 1'=-1:x N ,~1 I l> Similarly, for the sample variance of X : I N 2 Var(X)=-1:(X,-1') N-I . ., ). and the sample covariance between two variables X and Y: 1 N _ _ Cov{X,Y)=-1:(X,-xXy;-Y) N-1t-' Example Continued » The covariance between X and Y is then given by a xr = [(-IOX-1)+(-5)(-3)+(O)(-I)+(5)(4)+(IOXI»)/4 = 13.75 l> Finally, the correlation between X and Yis Pxr = 13.75/[(62.5)(7))112 = .657 Example Continued l> From Result 3, the mean of a sum is just the sum of the means: E(Z) = E(2X + 3 Y) = 2E(X) + 3E(Y) Since we know the mean of X and the mean of Y, E(Z) = 2(0) + 3(2) = 6
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KIC000015 - Example l> Example Continued Suppose we have...

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