which one has a lower covariance

Question

For this problem, I know that the mean of both plans are the same, so the only problem is which plan has a lower covariance. I know the following formulas: , but I don't know the value of Cor(X,Y). I only know it is between -1 and 1. So how am I supposed to find which plan has a lower variance? Thanks.

Answer 1

As you said, the expected value of both plans will be the same. Now let's look at the variance:

Plan 1: $Var(2X + 2Y) = 4Var(X) + 4Var(Y) + 8Cov(X,Y)$

Plan 2: $Var(3X + Y) = 9Var(X) + Var(Y) + 6Cov(X,Y)$

Using $Var(X) = Var(Y) = \sigma^2$, we have:

Plan 1: $Var(2X + 2Y) = 8\sigma^2 + 8Cov(X,Y)$

Plan 2: $Var(3X + Y) = 10\sigma^2 + 6Cov(X,Y)$

Looking at the difference, we find that we would prefer plan 1 over plan 2 when $Cov(X,Y) < \sigma^2$.

Answer 2

You have $\mathsf {Var}(aX+bY)=(a^2+b^2)\sigma^2 + 2ab\sigma^2\rho$ for some unknowns $\sigma^2$ and $\rho$ and two cases of the constants $a,b$, being: $(2,2)$ and $(3,1)$.

So you have been asked: What values of $\sigma^2,\rho$, would make $\mathsf {Var}(2X+3Y)>\mathsf {Var}(3X+Y)$, and which do the opposite?

Clearly $\sigma^2$ is not going to influence the outcome, so it is down to: what value of $\rho$ ?