This article develops an extension of the Rayleigh distribution with two parameters and greater flexibility which is an improvement over Lindley distribution, Rayleigh distribution and other generalizations of the Rayleigh distribution. The new model is known as “odd Lindley-Rayleigh Distribution”. The definitions of its probability density function and cumulative distribution function using the odd Lindley-G family of distributions are provided. Some properties of the new distribution are also derived and studied in this article with applications and discussions. The estimation of the unknown parameters of the proposed distribution is also carried out using the method of maximum likelihood. The performance of the proposed probability distribution is compared to some other generalizations of the Rayleigh distribution using three simulated datasets and a real life dataset. The results obtained are compared using the values of some information criteria evaluated with the parameter estimates of the fitted distributions based on the four datasets and it is revealed that the proposed distribution outperforms all the other fitted distributions. This performance has shown that the odd Lindley-G family of distribution is an adequate generator of probability models and that the odd Linley-Rayleigh distribution is a very flexible distribution for fitting different kinds of datasets better than the other generalizations of the Rayleigh distribution considered in this study.

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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)