Generating random values from non-normal and correlated distributions

I have a random variable X that is a mixture of a binomial and two normals (see what the probability density function would look like (first chart))

and I have another random variable Y of similar shape but with different values for each normally distributed side.

X and Y are also correlated, here's an example of data that could 开发者_如何学编程be plausible :

    X     Y
1.  0    -20
2. -5     2
3. -30    6
4.  7    -2
5.  7     2

As you can see, that was simply to represent that my random variables are either a small positive (often) or a large negative (rare) and have a certain covariance.

My problem is : I would like to be able to sample correlated and random values from these two distributions.

I could use Cholesky decomposition for generating correlated normally distributed random variables, but the random variables we are talking here are not normal but rather a mixture of a binomial and two normals.

Many thanks!

Note, you don't have a mixture of a binomial and two normals, but rather a mixture of two normals. Even though for some reason in your previous post you did not want to use a two-step generation process (first genreate a Bernoulli variable telling which component to sample from, and then sampling from that component), that is typically what you would want to do with a mixture distribution. This process naturally generalizes to a mixture of two bivariate normal distributions: first pick a component, and then generate a pair of correlated normal values. Your description does not make it clear whether you are fitting some data with this distribution, or just trying to simulate such a distribution - the difficulty of getting the covariance matrices for the two components will depend on your situation.