Questions on the Hypergeometric Distribution

Question

I'm considering a variable $X \sim HG(n,N,r)$ and I would like to solve for its expected value by breaking up $X$ into a sum of $r$ independent random variables $X_1,X_2,...,X_r.$ However, I am having issues figuring out this particular distribution that makes it such that $X = \sum_{i=1}^r X_i.$ I have a feeling that this is based on the successes of the $r$ available that show up in my sample of size $n$, but I am not sure if this is the appropriate way to move. Any suggestions?