How wil the parameters of a bi-variate normal distribution change if I rotate the x-y panel?

Question

For a bi-variate normal distribution:

If rho = 0, then the plot in x-y panel would look like:

I'm wonder, if I rotate the x-y panel a certain degree, into u-v panel, like this:

Then, what woudl the parameters of the bi-variate normal distributions in the new u-v panel be? i.e the sigma1, sigma2, rho ...

My thinking:

1. The mean value of the new bi-var Normal distribution would be easy to obtain, it's maybe something like

2. The main problem is the std, i.e. sigma1, sigma2, rho(or the covariance matrix). I don't know how they would change.

Answer 1

Denote the rotation matrix by $A$ and the covariance matrix by $R$. Further define a new random vector $w=Ax$ so:

1. Mean: $E[w] = E[Ax] = AE[x]$. Same result you got.
   2. Covariance matrix: $Cov[w]=Cov[Ax] = ACov[x]A^t=ARA^t$

In your case A is the rotation matrix.