Variance of two random variables given their variances and covariance.

Question

Y and Z are two random variables. Suppose Var(Y ) = 4, Var(Z) = 16, and Cov(Y, Z) = 3.

What is Var(3Z − 2Y )?

Answer 1

If you're learning from first principles: $$\begin{align}\mathsf{Var}(3Y-2Z) &= \mathsf E((3Y-2Z)^2)-\mathsf E(3Y-2Z)^2 \\ & \vdots \end{align}$$

Expand using Linearity of expectation, then use $\mathsf {Var}(Y)=\mathsf E(Y^2)-\mathsf E(Y)^2$ and $\mathsf {Cov}(Y,Z) = \mathsf E(YZ)-\mathsf E(Y)\mathsf E(Z)$, &c. to rearrange in terms of known values.

This will produce a rather elegant formula.

$\mathsf {Var}(3Y-2Z) = 9\,\mathsf {Var}(Y) - 12\,\mathsf {Cov}(Y,Z)+4\,\mathsf {Var}(Z)$