Some Instructional Issues in Hypergeometric Distribution

A brief introduction to sampling without replacement is presented. We represent the probability of a sample outcome in sampling without replacement from a finite population by three equivalent forms involving permutation and combination. Then it is used to calculate the probability of any number of successes in a given sample. The resulting forms are equivalent to the well known mass function of the hypergeometric distribution. Vandermonde’s identity readily justifies different forms of the mass function. One of the new form of the mass function embodies binomial coefficient showing much resemblance to that of binomial distribution. It also yields some interesting identities. Some other related issues are discussed

A mass function based on correlation coefficient and its application

A discrete distribution involving a product of two gamma functions has been proposed. It arises naturally in connection with the distribution of sample correlation coefficient based on a bivariate normal population. The first four factorial moments, raw moments, three corrected moments, coefficient of skewness and kurtosis have been derived. Some illustrations have been provided to show how the product moments of sample variances and correlation can be derived by exploiting the moments of the new distribution. These product moments are important for correlation analysis, covariance analysis, intra-sire regression and inference in bivariate normal population with a common mean.

Cardinality of Joint and Disjoint Sets and a Related Discrete Distribution

The number of elements belonging to the intersection of some overlapped sets and that belonging precisely to the junction of some overlapped sets are related. We have demonstrated a more straightforward presentation with some examples for a broad spectrum of students and instructors. It shows an apparent relationship between the cardinality of joint and disjoint sets. There is a probability distribution associated with it, and it is unexplored. We have provided some examples to illustrate it. We have derived the factorial moment structure of the probability distribution for the first time, and they found it to be elegant. We have also derived raw and corrected moments of the distribution

The distribution of a linear combination of two correlated chi-square variables

La distribución de una combinación lineal de dos variables chi cuadradoes conocida si las variables son independientes. En este artículo, se deriva ladistribución de una combinación lineal positiva de dos variables chi cuadradocuando estas están correlacionadas a través de una distribución chi cuadradobivariada. Algunas propiedades de esta distribución como la función característica,la función de distribución acumulada, sus momentos, momentoscentrados alrededor de la media, los coeficientes de sesgo y curtosis sonderivados. Los resultados coinciden con el caso independiente cuando lasvariables son no correlacionadas. La gráfica de la función de densidad estambién presentada

Moments of the product and ratio of two correlated chi-square variables

Bivariate chi-square distribution, Moments, Product of correlated chi-square variables, Ratio of correlated chi-square variables, 62E15, 60E05, 60E10,

Some useful integrals and their applications in correlation analysis

Correlation coefficient, Double integrals, Elliptical distribution, Bivariate t-distribution,