I want to generate a random matrix of size m x n as follows: for each i between 1 and m, take a random p_i number according to some fixed distribution (for example, uniform distribution in [0,10]). These numbers p_i are independent. Then I'll form a random matrix whose (i,j)-entry is chosen according to a probability distribution that depends on the parameter p_i (for example, it could be Poisson(p_i)). So entries in the same row are independent.
How can I do that?