Re: Does linearly dependent imply statistically dependent?

In article <1115232181.418018.175160@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<fjblurt@xxxxxxxxx> wrote:
>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.

.... but not "only if". Why don't you use an actual definition?

>> Does this imply that X, Y and Z are also statistically dependent?

Not quite. You could have a "trivial" case where (in the real definition)
one or (in your definition) two or three of X, Y, Z are identically 0.

>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.

Who said those variances and covariances exist?

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

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