Hypergeometric Distribution Variance via Indicator Covariances

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Joint Indicators / Covariance.

Your Task
An urn contains $N$ balls, of which $K$ are green and $N-K$ are red. You sample $n$ balls without replacement ($n \le N$). Let $X$ be the number of green balls drawn.
(1) Using indicator variables, compute $\mathbb{E}[X]$ and $\text{Var}(X)$.
(2) Identify the finite population correction factor.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.