Say I have the following : $$ (X, Y) \sim N_2(\mu, \Sigma) $$
Then what would be the distribution of $(2X,2Y)$ ?
Let $\Sigma = \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2\\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{pmatrix}$
I think $\rho$ shouldn't change since : $$ corr(ππ+π,ππ+π)=corr(π,π) $$ and that I should use the univariate transformation for the sigmas, so it would be : $$ (2X, 2Y) \sim N_2(2\mu, 4\Sigma) $$ but can't prove it, and my simulations don't seem to back it. Does anybody have a clue about this ? I can't find anything on the internet about multiplying a bivariate gaussian random variable with a constant