# Variance of a hypergeometric distribution

I'm trying to answer the following question from Ross's book:

A pond contains 100 fish, of which 30 are carp. If 20 fish are caught, what are the mean and variance of the number of carp among the 20? What assumptions are you making?

Letting $$X$$=total number of carp caught, I know the assumption is that X is a hypergeometric RV. Letting $$X_i$$=1 if the $$i^{th}$$ carp is caught, I've gotten this far:

$$Var[X]=Var[\sum_{i=1}^{20} X_i] = \sum_{i=1}^{30} Var[X_i] + 2\sum_{i

I understand what all this represents, but I just don't know how to actually evaluate the summations to get a final number.

• Why not begin with the definition of variance in terms of the first two moments of the distribution? You can compute those moments with relative ease.
– whuber
Commented May 8, 2019 at 16:26
• My professor skipped the section on moments, so I'm not familiar with them. Is that the only way to evaluate this? Commented May 8, 2019 at 16:29
• There are (as usual) a very large number of ways to get an answer, but the way you're proceeding seems rather cumbersome. There are clever ways to determine the covariances--the problem is that they are nonzero--but such methods require insight or experience, whereas the calculation from the definitions requires only basic skills with sums.
– whuber
Commented May 8, 2019 at 16:41