论文标题
一般超小几幅分布的平均方差和渐近特性
Mean, Variance and Asymptotic Property for General Hypergeometric Distribution
论文作者
论文摘要
一般超几何分布（GHGD）定义：从包含$ n $元素的有限空间$ n $，随机选择$ t $ subsets $ m_i $（每个包含$ m_i $元素，$ 1 \ geq i \ geq t $），确切的$ x $ elements的可能性是$ x $ timpy $ t $ time $ t $ t $ t $ t $ t $ t} $）？ GHGD描述了随机变量的分布$ x_t $和$ x _ {\ geq t} $。在我们之前的结果中，我们获得了特殊情况（$ t \ leq 7 $）的数学期望和差异的公式，但未提供证明。在这里，我们在任何情况下完成了$ x_t $和$ x _ {\ geq t} $的均值和差异的确切公式，并提供了严格的数学证明。另外，我们给出了变量的渐近特性。当平均值接近0时，方差快速接近平均值的值，实际上，它们的差异是均值的高阶无限量。因此，当平均值足够小（$ <1 $）时，它可以用作相当准确的方差近似值。
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.
