Mean, Variance and Asymptotic Property for General Hypergeometric Distribution

TL;DR

This paper derives exact formulas and proofs for the mean and variance of variables in the general hypergeometric distribution, and analyzes their asymptotic behavior, especially when the mean is small.

Contribution

It provides the first complete formulas and rigorous proofs for mean and variance of GHGD variables for any case, extending previous partial results.

Findings

Exact formulas for mean and variance for any GHGD case

Asymptotic behavior shows variance approximates mean when mean is small

When mean < 1, variance can be accurately approximated by the mean

Abstract

General hypergeometric distribution (GHGD) definition: from a finite space $N$ containing $n$ elements, randomly select totally $T$ subsets $M_i$ (each contains $m_i$ elements, $1 \geq i \geq T$), what is the probability that exactly $x$ elements are overlapped exactly $t$ times or at least $t$ times ($x_t$ or $x_{\geq t}$)? The GHGD described the distribution of random variables $x_t$ and $x_{\geq t}$. In our previous results, we obtained the formulas of mathematical expectation and variance for special situations ($T \leq 7$), and not provided proofs. Here, we completed the exact formulas of mean and variance for $x_t$ and $x_{\geq t}$ for any situation, and provided strict mathematical proofs. In addition, we give the asymptotic property of the variables. When the mean approaches to 0, the variance fast approaches to the value of mean, and actually, their difference is a higher order infinitesimal of mean. Therefore, when the mean is small enough ($<1$), it can be used as a fairly accurate approximation of variance.