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Definition of Covariance

Definition of Covariance. The covariance of X & Y , denoted Cov ( X , Y ), is the number where m X = E ( X ) and m Y = E ( Y ). Computational Formula:. Variance of a Sum. Covariance and Independence. If X & Y are independent, then Cov ( X , Y ) = 0.

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Definition of Covariance

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  1. Definition of Covariance • The covariance of X & Y, denoted Cov(X,Y), is the numberwhere mX = E(X) and mY = E(Y). • Computational Formula:

  2. Variance of a Sum

  3. Covariance and Independence • If X & Y are independent, then Cov(X,Y) = 0. • If Cov(X,Y) = 0, it is not necessarily true that X & Y are independent!

  4. The Sign of Covariance • If the sign of Cov(X,Y) is positive, above-average values of X tend to be associated with above-average values of Y and below-average values of X tend to be associated with below-average values of Y. • If the sign of Cov(X,Y) is negative, above-average values of X tend to be associated with below-average values of Y and vice versa. • If the Cov(X,Y) is zero, no such association exists between the variables X and Y.

  5. Correlation • The sign of the covariance has a nice interpretation, but its magnitude is more difficult to interpret. • It is easier to interpret the correlation of X and Y. • Correlation is a kind of standardized covariance, and

  6. Conditions for X & Y to be Uncorrelated • The following conditions are equivalent:Corr(X,Y) = 0Cov(X,Y) = 0E(XY) = E(X)E(Y)in which case X and Y are uncorrelated. • Independent variables are uncorrelated. • Uncorrelated variables are not necessarily independent!

  7. Let (X, Y) have uniform distribution on the four points (-1,0), (0,1), (0,-1) and (1,0). Show that X and Y are uncorrelated but not independent. • What is the variance of X + Y?

  8. Let T1 and T3 be the times of the first and third arrivals in a Poisson process with rate l. • Find Corr(T1,T3). • What is the variance of T1 + T3?

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