I'm having trouble approaching this question as I am unsure as to how certain aspects come into play. The question is as follows:
Let $X$ and $Y$ be jointly normal random variables with means $E[X] = 0$ and $E[Y] = -1$, variances $Var[X] = 4$ and $Var[Y] = 9$. Define a random variable $Z = 2X + 3Y$.
(a) Assume that $X$ and $Y$ are uncorrelated. Find the PDF of $Z$, $f_Z(z)$ for $-\infty < z < \infty$.
(b) Assume that $X$ and $Y$ have correlation coefficient $\rho = -0.4$. Find the PDF of $Z$, $f_Z(z)$ for $-\infty < z < \infty$.
I am not entirely sure what relevance correlation has, as external sources do not seem to mention how they may affect the PDF (http://www.stat.ucla.edu/~nchristo/introstatistics/introstats_normal_linear_combinations.pdf).
So far I have tried to solve (a) by considering $2X + 3Y \sim N(2\times0 + 3\times (-1), \sqrt{2^2 \times 4 + 3^2 \times 9}$ and thus, $f_Z(z) = \frac{1}{\sqrt{194\pi}}\text{exp}\left({-\frac{1}{2}\left(\frac{x+3}{\sqrt{97}}\right)^2}\right)$, but again, I have no idea if this is correct.
Any help would be greatly appreciated.
If normal random variables $X$ and $Y$ are independent, then $W = 2X + 3Y$ has $$E(W)= E(2X+3Y) = 2E(X)+3E(Y) = 2(0) + 3(-1) = -3.$$ Also, $$Var(2X+3Y) = 2^2Var(X) + 3^2Var(Y).$$ Furthermore, $W = 2X+3Y$ is normal.
However, if $X$ and $Y$ are correlated, then the equation above for the variance $W$ does not hold. It is replaced by a different equation that takes covariance into account (and you can get covariance from correlation and variances.)