Multivariate Normal distribution

https://www0.maths.ox.ac.uk/system/files/coursematerial/2015/3188/24/sheet6.pdf

Question 5i).

So basically i have no idea how to start off. I tried calculating the probability density for X = (X1, X2)^T , but the numbers do not work out nicely at all.

I have also found that E(Y) = 0 and Var(Y) = 172 (as 4*25 + 9*16 - 3*2*12), but don't know what to do with this.

pls help xx

@Gregorius

The big theorem here that helps you out is that if $X_1$ and $Y_2$ are normally distributed, then so is $Y = aX_1 + bX_2$. All you need to find out is the mean and the variance of Y and that uniquely specifies the distribution of Y.

You've found the mean of Y, but your calculation of the variance of Y is a bit off. Remember that

$\displaystyle Var(Y) = Var(2X_1) + Var(3X_2) + 2 Cov(2X_1, -3X_2) = 4 \times Var(X_1) + 9 \times Var(X_2) - 2 \times 2 \times 3 \times Cov(X_1, X_2)$

which gives you much nicer numbers!

Argh i forgot the 2. Thanks