Bivariate normal distribution proving and finding parameters

Question

I have $X$ and $Y$ which are independent random variables following the normal distribution.

How should I prove that a random variable ($Y-2X$, $X+3Y$) has bivariate normal distribution?

And how should I find the values of parameters for it?

Thanks in advance!

Answer 1

You don't prove that they follow a birvariate normal. When you pair normal distributions like that, they follow a bivariate normal.Let $S = Y-2X$ and $T = X+3Y$. Then the parameters of the bivariate normal are the expectation of $S$, $\mu_S$, the expectation of $T$, $\mu_T$, the variances of $S$ and $T$, $\sigma_S^2,\sigma_T^2$, and the covariance of (S,T), called it $\rho$.