Find the covariance of $X$ and $Y$ given $Z = 3x - 2y$ and the standard deviation of $x, y$ and $z$.

Question

This is the exact problem:

Suppose that $X, Y$ are random variables with $Sx =2, Sy = 3$. Let $Z = 3X - 2Y$, and assume that $Sz = 6$. Find the covariance, $\text{cov}(X, Y)$.

I have equations for covariance but they involve the means of $X$ and $Y$, or the correlation of $X$ and $Y$.

Any ideas?

Answer 1

We work with variances, not standard deviations. We have $$\text{Var}(Z)=\text{Var}(3X-2Y)=9\text{Var}(X)+4\text{Var}(Y)-12\text{Cov}(X,Y).\tag{1}$$ Formula (1) together with the given information are enough to calculate $\text{Cov}(X,Y)$.

Remark: Although the standard defining formulas for variance and covariance involve the mean, in a sense the mean is irrelevant. If $X'=a+X$ and $Y'=b+Y$, then the variance of $X'$ is the same as the variance of $X$, the variance of $Y'$ is the same as the variance of $Y$, and the covariances also match.

Answer 2

The trick is to recall the formula $$\operatorname{Var}[cA + dB] = c^2 \operatorname{Var}[A] + d^2 \operatorname{Var}[B] + 2cd \operatorname{Cov}[A,B].$$ Apply this for a suitable choice of constants $c, d$ and random variables $A, B$.