Similar to the question I asked before, with one subtle difference.
If $X \sim N(20,2^2)$ and $Y \sim N(10,1)$ and $X$ and $Y$ are correlated with $\rho = 0.5$
then find:
$a)$ the covariance between $X$ and $Y$
$b)$ The distribution of the vector $(X,Y)$
$c)$ give the distribution of $U = 2X-3Y$ and specify the mean and variance
For $a) $ $\rho = COV(X,Y)/ \sqrt{VAR(X)VAR(Y)} $ so $COV(X,Y) = 1$
for
$b)$
$(X,Y) \sim N(\mu, \Sigma) = \sim N[ \begin{pmatrix} 20 \\ 10 \end{pmatrix}, \begin{pmatrix} 4 & 0.5 \\ 0.5 & 1 \end{pmatrix}]$
for the next part I am not entirely sure:
$c)$
Since expectation is a linear operator: $E(U) = E(2X) + E(-3Y) = 2E(X) - 3E(Y) = 2(20) - 3(10) = 10$
Now for Variance I am a little unsure,
$VAR(2X-3Y) = VAR(2X) + 2COV(2X,-3Y) + VAR(3Y)$
$= 4VAR(X) - 12COV(X,Y) + VAR(3Y)$
$= 4(4) - 12 + 9(1) = 13$
And the distribution of $2X-3Y$ is $U \sim N(10,13)$
Are my solutions correct?