Python generate 3 correlated array of random noise with different distributions

Question

I need to generate random noise for 3 variables, i.e. X, Y and Z. I need X to be lognormal, and Y and Z is normal. The correlation matrix should be:

1, -0.8, 0.5
-, 1,   -0.5
0.5,-0.5,  1

The std of X and Y and Z should be 0.5, 0.3 and 20. The mean of X and Y should be 1, and 0 for Z. How can I do that?

I know I can use Generate correlated data in Python (3.3) to generate 3 normally distributed. So how can I apply it to my case?

I think I can firstly generate X1=log(X), Y and Z at first. But how can I set the correlation matrix? i.e. if the corr(X, Y)=-0.8, what is corr(log(X), Y)?

-----advanced version

what if I want X and Y and Z all be AR(1) process?

Thanks!