Question: Let U be a random 31 vector. Show that cov(U ) is non-negative definite, and positive definite unless there is a 31 fixed (i.e., nonrandom)

Let U be a random 3×1 vector. Show that cov(U ) is non-negative definite, and positive definite unless there is a 3×1 fixed (i.e., nonrandom) vector such that c

U = c

E(U ) with probability 1. Hints. Can you compute var(c

U ) from cov(U )? If that hint isn’t enough, try the case E(U ) = 03×1. Comment: if c U = c E(U ) with probability 1, then U − E(U ) concentrates in a fixed hyperplane.

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