Expectation Identity of the Hypergeometric Distribution and Its Application in the Calculations of High-Order Origin Moments

We provide a novel method to analytically calculate the high-order origin moments of a hypergeometric distribution, that is, the expectation identity method. First, the expectation identity of the hypergeometric distribution is discovered and summarized in a theorem. After that, we analytically calculate the first four origin moments of the hypergeometric distribution by using the expectation identity. Furthermore, we analytically calculate the general kth (k=1,2,…) origin moment of the hypergeometric distribution by using the expectation identity, and the results are summarized in a theorem. Moreover, we use the general kth origin moment to validate the first four origin moments of the hypergeometric distribution. Next, the coefficients of the first ten origin moments of the hypergeometric distribution are summarized in a table containing Stirling numbers of the second kind. Moreover, the general kth origin moment of the hypergeometric distribution by using the expectation identity is restated by another theorem involving Stirling numbers of the second kind. Finally, we provide some numerical and theoretical results.

Estimating COVID-19 cases infected with the variant alpha (VOC 202012/01): an analysis of screening data in Tokyo, January-March 2021

Background In Japan, a part of confirmed patients’ samples have been screened for the variant of concern (VOC), including the variant alpha with N501Y mutation. The present study aimed to estimate the actual number of cases with variant alpha and reconstruct the epidemiological dynamics. Methods The number of cases with variant alpha out of all PCR confirmed cases was estimated, employing a hypergeometric distribution. An exponential growth model was fitted to the growth data of variant alpha cases over fourteen weeks in Tokyo. Results The weekly incidence with variant alpha from 18–24 January 2021 was estimated at 4.2 (95% confidence interval (CI): 0.7, 44.0) cases. The expected incidence in early May ranged from 420–1120 cases per week, and the reproduction number of variant alpha was on the order of 1.5 even under the restriction of contact from January-March, 2021, Tokyo. Conclusions The variant alpha was predicted to swiftly dominate COVID-19 cases in Tokyo, and this has actually occurred by May 2021. Devising the proposed method, any country or location can interpret the virological sampling data.

APPLICATIONS OF TWO NON-CENTRAL HYPERGEOMETRIC DISTRIBUTIONS OF BIASED SAMPLING STATISTICAL MODELS

Statistical models of biased sampling of two non-central hypergeometric distributions Wallenius' and Fisher's distribution has been extensively used in the literature, however, not many of the logic of hypergeometric distribution have been investigated by different techniques. This research work examined the procedure of the two non-central hypergeometric distributions and investigates the statistical properties which includes the mean and variance that were obtained. The parameters of the distribution were estimated using the direct inversion method of hyper simulation of biased urn model in the environment of R statistical software, with varying odd ratios (w) and group sizes (mi). It was discovered that the two non - central hypergeometric are approximately equal in mean, variance and coefficient of variation and differ as odds ratios (w) becomes higher and differ from the central hypergeometric distribution with ω = 1. Furthermore, in univariate situation we observed that Fisher distribution at (ω = 0.2, 0.5, 0.7, 0.9) is more consistent than Wallenius distribution, although central hypergeometric is more consistent than any of them. Also, in multinomial situation, it was observed that Fisher distribution is more consistent at (ω = 0.2, 0.5), Wallenius distribution at (ω = 0.7, 0.9) and central hypergeometric at (ω = 0.2)

Refining One Theorem For The Romanovsky Distribution

The paper considered a refinement of the theorem for a negative-hypergeometric distribution( the Romanovsky distribution), i.e., convergence over variation of the Romanovsky distribution by Erlang distributions. The theorem is proved by the direct asymptotic method.

Order-Preserving and Efficient One-to-Many Search on Encrypted Data

Order-preserving encryption (OPE) is an useful tool in cloud computing as it allows untrustworthy server to execute range query or exact keyword search directly on the ciphertexts. It only requires sub-linear time in the data size while the queries are occurred. This advantage is very suitable in the cloud where the data volume is huge. However, the order-preserving encryption is deterministic and it leaks the plaintexts’ order and its distribution. In this paper, we propose an one-to-many OPE by taking into account the security and the efficiency. For a given plaintext, the encryption algorithm firstly determines the corresponding ciphertext gap by performing binary search on ciphertext space and plaintext space at the same time. An exact sample algorithm for negative hypergeometric distribution is used to fix the size of the gap. Lastly a value in the gap is randomly chosen as the mapping of the given plaintext. It is proven that our scheme is more secure than deterministic OPE with realizing efficient search. In particular, a practical and exact sampling algorithm for the negative hypergeometric distribution (NHGD) is first proposed.

Generalized hypergeometric distribution and its applications on univalent functions

The purpose of the present paper is to introduce a generalized hypergeometric distribution and obtain some necessary and sufficient conditions for generalized hypergeometric distribution series belonging to certain classes of univalent functions associated with the conic domains. We also investigate some inclusion relations. Finally, we discuss an integral operator related to this series.