Unformatted text preview: pairwise uncorrelated.
That is, there is no diﬀerence between a set of random variables being uncorrelated or being pairwise
uncorrelated. Recall from Section 2.4.1 that, in contrast, independence of three or more events is
a stronger property than pairwise independence. Therefore, mutual independence of n random
variables is a stronger property than pairwise independence. Pairwise independence of n random
variables implies that they are uncorrelated.
Covariance is linear in each of its two arguments, and adding a constant to a random variable
does not change the covariance of that random variable with other random variables:
Cov(X + Y, U + V ) = Cov(X, U ) + Cov(X, V ) + Cov(Y, U ) + Cov(Y, V )
Cov(aX + b, cY + d) = acCov(X, Y ), 152 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES for constants a, b, c, d. The variance of the sum of uncorrelated random variables is equal to the
sum of the variances of the random variables. For example, if X and Y are uncorrelated,
Var(X + Y ) = Cov(X + Y, X + Y ) = Cov(X, X ) + Cov(Y, Y ) + 2Cov(X, Y ) = Var(X ) +...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.
- Spring '08
- The Land