Random vectors and multivariate distributions

We now extend the concept of random variables to treat random vectors, for which we need a number of additional definitions. 

Definition Covariance and correlation of two random variables hetX and Fbe two random variables. The covariance of AT and Y is 

A related concept is the correlation ofXandy,corr(X, Y) = cov(A, Y)/. vax X) + var(F). IfX and Y are independent, cov(A, F) = 0 however, a covariance of zero does not necessarily imply that the two variables must be independent (although it suggests that they are). If cov(X, F) 0, then when A is greater than its mean E X), Y tends also to be greater than its mean (F). Conversely, if cov(A, F) 0, then if A E X), it is more probable that F is less than E Y). A nonzero covariance means only that the two variables tend to behave in a related manner, it does not mean that there is a cause and effect relationship among them. Asserting the latter is a common fallacy. 

Let n be a vector whose components are random variables, not necessarily independent. Then, the covariance matrix of v, cov(n), has elements 

If in addition, each component of v has the same variance then 

---