Mohammad Shakil

Normal and Student's t Distributions and Their Applications

The most important properties of normal and Student t-distributions are presented. A number of applications of these properties are demonstrated. New related results dealing with the distributions of the sum, product and ratio of the independent normal and Student distributions are presented. The materials will be useful to the advanced undergraduate and graduate students and practitioners in the various fields of science and engineering.

An introduction to order statistics

Basic definitions.- Distributions of order statistics.- Sample quantiles and ranges.- Representations for order statistics.- Conditional distributions of order statistics.-Order statistics for discrete distributions.- Moments of order statistics: general relations.- Moments of uniform and exponential order statistics.- Moment relations for order statistics: normal distribution.- Asymptotic behavior of the middle and intermediate order statistics.- Asymptotic behavior of the extreme order statistics.- Some properties of estimators based on order statistics.- Minimum variance linear unbiased estimators.- Minimum variance linear unbiased estimators and predictors based on censored samples.- Estimation of parameters based on fixed number of sample quantiles.- Order statistics from extended samples.- Order statistics and record values.- Characterizations of distributions based on properties of order statistics.- Order statistics and record values based on Falpha distributions.- Generalized order statistics.- Compliments and problems.

A NOTE ON THE CHARACTERIZATIONS OF PARETO DISTRIBUTION BY UPPER RECORD VALUES

Many researchers have studied the characterizations of prob- ability distributions based on record values. It appears from literature that not much attention has been paid to the characterizations of the Pareto distribution. In this note, some new results on the characteriza- tions of the Pareto distribution by upper record values have been estab- lished.

A Note on a Characterization of Gompertz-Verhulst Distribution

Characterization of a probability distribution plays an important role in probability and statistics. This paper considers a new characterization of Gompertz-Verhulst distribution. It is hoped that the findings of the paper will be useful for researchers in different fields of applied sciences.

On the Characterizations of Chenrs Two-Parameter Exponential Power Life-Testing Distribution

Characterizations of probability distributions play important roles in probability and statistics. Before a particular probability distribution model is applied to fit the real world data, it is essential to confirm whether the given probability distribution satisfies the underlying requirements by its characterization. A probability distribution can be characterized through various methods. In this paper, we provide the characterizations of Chen’s two-parameter exponential power life-testing distribution by truncated moment. 2010 Mathematics Subject Classifications: 60E05, 62E10, 62E15, 62G30

Sum, Product and Ratio for the Student’s t Random Variables

This chapter presents the distributions of the sum \(X + Y\), product XY, and ratio X/Y when X and Y are independent random variables and have the Student’s t distributions with appropriate degrees of freedoms.

Concluding Remarks and Some Future Research

The normal and Student’s distributions are two of the most important distributions in statistics. This book has reviewed the normal and Student’s distributions, and their applications.

Sum, Product and Ratio for the Normal Random Variables

The distributions of the sum, product, and ratio of two independent random variables arise in many fields of research, for example, automation, biology, computer science, control theory, economics, engineering, fuzzy systems, genetics, hydrology, medicine, neuroscience, number theory, statistics, physics, psychology, reliability, risk management, etc. (for details, see Grubel (1968), Rokeach and Kliejunas (1972), Springer (1979), Kordonski and Gertsbakh (1995), Ladekarl et al. (1997), Amari and Misra (1997), Sornette (1998), Cigizoglu and Bayazit (2000), Brody et al. (2002), Galambos and Simonelli (2005), among others).

Sum, Product and Ratio for the Normal and Student’s t Random Variables

The distributions of the sum \(X+Y\), product \(XY\), and ratio \(X/Y\), when \(X\) and \(Y\) are independent random variables and belong to different families, are of considerable importance and current interest. These have been recently studied by many researchers, (among them, Nadarajah (2005b, c, d) for the linear combination, product and ratio of normal and logistic random variables, Nadarajah and Kotz (2005c) for the linear combination of exponential and gamma random variables, Nadarajah and Kotz (2006d) for the linear combination of logistic and Gumbel random variables, Nadarajah and Kibria (2006) for the linear combination of exponential and Rayleigh random variables, Nadarajah and Ali (2004) for the distributions of the product \(XY\) when \(X\) and \(Y\) are independent Laplace and Bessel random variables respectively, Ali and Nadarajah (2004) for the product and the ratio of t and logistic random variables, Ali and Nadarajah (2005) for the product and ratio of t and Laplace random variables, Nadarajah and Kotz (2005b) for the ratio of Pearson type VII and Bessel random variables, Nadarajah (2005c) for the product and ratio of Laplace and Bessel random variables, Nadarajah and Ali (2005) for the distributions of \(XY\) and \(X/Y\), when \(X\) and \(Y\) are independent Student’s and Laplace random variables respectively, Nadarajah and Kotz (2005a) for the product and ratio of Pearson type VII and Laplace random variables, Nadarajah and Kotz (2006a) for the product and ratio of gamma and Weibull random variables, Shakil, Kibria and Singh (2006) for the ratio of Maxwell and Rice random variables, Shakil and Kibria (2007) for the ratio of Gamma and Rayleigh random variables, Shakil, Kibria and Chang (2007) for the product and ratio of Maxwell and Rayleigh random variables, and Shakil and Kibria (2007) for the product of Maxwell and Rice random variables, are notable). This chapter studies the distributions of the sum \(X+Y\), product \(XY\), and ratio \(X/Y\), when \(X\) and \(Y\) are independent normal and Student’s t random variables respectively.

Product of the Normal and Student’s t Densities

The distributions of the product of two random variables have a great importance in many areas of research both from the theoretical and applications point of view.