I have a question about going from the distribution of a single variable to an iid sample.
As far as I can tell, if I have a r.v. that is normally distributed and a sample of iid r.v.'s, I then determine the joint distribution of the variables via the product of the densities.
When I look at a r.v. that is Bernoulli, then the iid sample is distributed as Binomial (which still makes somewhat sense to me).
I'm not sure how I get from a r.v. that is Poisson(l) to an iid sample that is distributed as Poisson(nl); specifically, it seems that I am looking at the summation of the r.v.'s rather than the product? Why or where am I not understanding this correctly?