Mean, Variance and Asymptotic Property for General Hypergeometric Distribution
share
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 ≥ i ≥ T), what is the probability that exactly x elements are overlapped exactly t times or at least t times (x_t or x_≥ t)? The GHGD described the distribution of random variables x_t and x_≥ t. In our previous results, we obtained the formulas of mathematical expectation and variance for special situations (T ≤ 7), and not provided proofs. Here, we completed the exact formulas of mean and variance for x_t and x_≥ 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.
READ FULL TEXT
