On lower bounds for hypergeometric tails

Abstract: Let $n,k$ be positive integers such that $n\geq k$, and let $H$ be a hypergeometric random variable counting the number of black marbles in a sample without replacement of size $k$ from an urn that contains $i\in {1,\ldots, n}$ black and $n - i$ white marbles. It is shown that [ \mathbb{P}(H \ge \mathbb{E}(H)) \ge k/n\, , \, \text{when} \,\, n\ge 8k \, . ] Furthermore, provided that $1\le \mathbb{E}(H)\le \min{i,k}-2$ as well as that $\frac{(n-i)(n-k)}{n}>1$, it is shown that [ \mathbb{P}(H\ge \mathbb{E}(H)) \,\ge\, \frac{e^{-1/8}}{4\sqrt{2}} \cdot \sqrt{\frac{n-1}{n}} \cdot\frac{ \sqrt{\text{Var}(H)} }{1 + \sqrt{1+ \frac{n-1}{n-k}\cdot\text{Var}(H)}}\, . ] Auxiliary results which may be of independent interest include an upper bound on the tail conditional expectation and a lower bound on the mean absolute deviation of the hypergeometric distribution.