Re: Does linearly dependent imply statistically dependent?

wangx...@xxxxxxxxx wrote:
> Given three random vectors X, Y, and Z.
> We say X, Y, Z are linearly dependent if X = aY + bZ, where a and b
are
> scalar numbers and at least one of them is not zero.
> Does this imply that X, Y and Z are also statistically dependent?
>
> Thanks a lot
> Sophia

Yes. If they were independent, we would have

0 = Cov(X,Y) = a Var(Y) + b Cov(Y,Z)
0 = Cov(X,Z) = a Cov(Y,Z) + b Var(Z)
0 = Cov(Y,Z)

so that a Var(Y) = b Var(Z) = 0. If we rule out the trivial case where
one of Y, Z is a.s. constant, they both have positive variance, and
then a=b=0.

.

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