Bivariate normal: Expected value and variance

Question

I have this simple exercise to do and I'm new to this topic.

Seeing the slides of my professor, I would solve the problem in this way:

$E(Y)=c_1 μ_1+c_2 μ_2$

$E(Y)=-54.2424$

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$Var(Y)=σ_{22}$

$Var(Y)=0.65$

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I don't have the solutions so I take this opportunity to know if my reasoning has errors or not. If there are, how should I proceed to solve it?

Answer 1

Your expression for $E(Y)$ is correct. The variance of $Y$ won't be $\sigma_{22}$ - that is the variance of $X_2$. For the variance of $Y$ use the equation

$$Var(Y) = Var(c_1 X_1 + c_2 X_2) = c_1^2 Var(X_1) + c_2^2 Var(X_2) + 2c_1 c_2 Cov(X_1,X_2)$$

which is the general equation for the variance of a linear combination of two random variables.