Suppose I want N samples of X and Y, which are two random variables which are correlated with a certain value of correlation coefficient. They have generally non-gaussian PDF, but at the moment let's assume they are equal (for instance assume X,Y are Beta).
How to find these samples? I know two methods:
Suppose I can generate N normally distributed random samples with known correlation coefficient (bivariate gaussian distribution). I take these two groups of samples and apply the gaussian CDF to get uniform random variables. Then, I apply to these samples the Beta ICDFs to provide the final samples of X and Y with Beta distribution. I have been told this method inherently utilizes a gaussian copula as the original coupling between the input random variables was that of a gaussian bivariate distribution.
This way I generate again 2 groups of N normally distributed random samples, but without correlation. I turn them to Beta the same way as in 1, but now these groups are uncorrelated. Now I multiply them for chol(Covariance Matrix). I get a couple of Beta random variables.
My questions are: a) Which is the inherent difference between these two methods? Are them inherently equivalent? I know that copula method can be adapted to generate any type of dependence (also in absence of correlation). Is this true also for the choleski approach?
b) Is the choleski method effective also for generated X and Y with different PDFs (for instance one is Beta and one is Poisson?)