Calculating covariance from variances

[gajohnson]

Homework Statement

Suppose that X1, X2, and X3
are independent random variables with variances 3, 4, and 8, respectively.
Let Y1 = 2X1 + 3X2, Y2 = X3 – X2, and Y3 = X1 + X2 + X3. (a) Using the general
relationship
Cov(W+X, Y+Z) = Cov(W,Y) + Cov(W, Z) + Cov(X, Y) + Cov(X, Z), find
Cov(Yi, Yj) for all i, j.

Homework Equations

The Attempt at a Solution

I can set up the Cov(Yi, Yj) for all i, j easily enough, but I do not understand how to calculate, say, 2Cov(X1, X3) just from the variances of X1 and X3. I know this is trivial, but any help would be greatly appreciated.

 

[Ray Vickson]

Homework Statement

Suppose that X1, X2, and X3
are independent random variables with variances 3, 4, and 8, respectively.
Let Y1 = 2X1 + 3X2, Y2 = X3 – X2, and Y3 = X1 + X2 + X3. (a) Using the general
relationship
Cov(W+X, Y+Z) = Cov(W,Y) + Cov(W, Z) + Cov(X, Y) + Cov(X, Z), find
Cov(Yi, Yj) for all i, j.

Homework Equations

The Attempt at a Solution

I can set up the Cov(Yi, Yj) for all i, j easily enough, but I do not understand how to calculate, say, 2Cov(X1, X3) just from the variances of X1 and X3. I know this is trivial, but any help would be greatly appreciated.

What is the covariance of two independent random variables?

 

[gajohnson]

What is the covariance of two independent random variables?

Oh, boy...I seem to have missed the "independent" condition in the problem. It works out nicely that each combination seems to have a random variable with a covariance of itself somewhere in there. The rest all becomes 0s.

Thanks!

 

[Ray Vickson]

b

Oh, boy...I seem to have missed the "independent" condition in the problem. It works out nicely that each combination seems to have a random variable with a covariance of itself somewhere in there. The rest all becomes 0s.

Thanks!

Also: to compute the covariances you need to know the means of the X_i, which seem to not have been given. Were they given, and you just forgot to include them here?

 

[gajohnson]

b

Also: to compute the covariances you need to know the means of the X_i, which seem to not have been given. Were they given, and you just forgot to include them here?

They were not given but, thankfully, because they are independent, the only covariances that are not 0 are those that are just a covariance with itself, i.e. the variance already given--usually multiplied by some constant.