Find the mean and variance - covariance matrix of the random vector

Question

Given random vector x = x1, x2, x3

ε = 25 -2 4

-2 4 1

4 1 9

find the mean and variance - covariance matrix of the random vector

y = y1

y2

y1 = x1 - 2x2 + 2x3

y2 = 3x1 + x2 - x3

Answer 1

The mean of x1, x2, and x3 are not shown, so the default assumption is that they have mean 0

Then, the mean of any linear combination of them is also 0.

If ε is the variance covariance matrix for x1, x2, x3, then, let Ax = y.

The variance - covariance matrix will be AεA^T

1 -2 2 25 -2 4 1 3 =

3 1 -1 -2 4 1 -2 1

4 1 9 2 -1

93 83

83 200