On lower bounds for hypergeometric tails

TL;DR

This paper establishes new lower bounds for the probability that a hypergeometric random variable exceeds its expectation, under specific conditions, with auxiliary results of independent interest.

Contribution

It introduces novel lower bounds for hypergeometric tail probabilities and provides auxiliary bounds on tail conditional expectation and mean absolute deviation.

Findings

Proves that \\mathbb{P}(H \\geq \\mathbb{E}(H)) \\geq k/n for n \\geq 8k.

Derives a refined lower bound involving variance and other parameters.

Provides auxiliary bounds on tail conditional expectation and mean absolute deviation.

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 $1\le \mathbb{E}(H)\le \min\{i,k\}-1$ as well as that $\frac{(n-i)(n-k)}{n}>1$, it is shown that \[ \mathbb{P}(H\ge \mathbb{E}(H)) \,\ge\, \frac{e^{-1/12}}{4\sqrt{2}} \cdot \sqrt{\frac{n-1}{n}} \cdot\frac{ \sqrt{\text{Var}(H)} }{1 + \sqrt{1+ \frac{n-1}{n-k}\cdot\text{Var}(H)}}\, . \] 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.