Calculating Mean and Covariance Matrix with New Variables?

[retspool]

My professor sucks
she hasnt gone over mean vector and she expects up to solve this

let z1, z2, z3 be the random variables with mean vector and covariance matrix given below

mean vector = [1 2 3]T. T = transpose

covariance vector

3 2 1
2 2 1
1 1 1


Define the new variables
y1 = z1 + 2z3; y2 = z1 + z2 - z3; y3 = 2z1 + z2 + z3 - 7
(a) Find the mean vector and the covariance matrix of (y1; y2; y3).
(b) Find the mean and variance of
y =(y1 + y2 + y3)/3

Thanks

 

[lanedance]

i assume you are talking about multivariate guassian distributed random variables, see below
http://en.wikipedia.org/wiki/Multivariate_normal_distribution

the means will sum directly, though you'll have to think a bit more about the covariances...

you could either consider each element of the covariance directly or you could write the sum oand try and manipulate it into the normal form

 

[statdad]

"she hasnt gone over mean vector and she expects up to solve this"

I'm skeptical of that comment.

You can write the new vector $$(y_1, y_2, y_3)'$$ as a linear transformation of the original variables, then apply the same transformation to the original mean vector.
There are general rules for transforming a covariance matrix (not covariance vector) from one set of variables to another - more matrix multiplication. The processes do not depend on the assumption of normality.

 

[lanedance]

good point on the normality assumption

see here for more on covariance matrices
http://en.wikipedia.org/wiki/Covariance_matrix