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Finding a variance, given a dispersion matrix

Q: Let A be a 3 x 1 vector of random variables such that:
D[A] = mat[5 2 3; 2 3 0; 3 0 2], where D[A] is the dispersion matrix of A.

Find the variance of Y = A1 - 2A2 + A3.

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A: Despite not knowing the elements of A, we can still find this variance. Here's how...

There is a Corollary to a Theorem on the topic of Covariance Operators that says that for any constant vector c, we have that,

D[c'X] = c'(D[X])c.

We can use this. Let's examine this a bit closer, with respect to ...

Solution Summary

A step-by-step solution shows how to find the variance of a random variable, Y, which happens to be a linear combination of unknown random variables, A=vec[A1;A2;A3], for which we only know D[A], the dispersion matrix of A.