This Book is the second edition that intended to be textbook studied for undergraduate/ postgraduate course in mathematical statistics. In order to achieve the goals of the book, it is divided into the following chapters. Chapter One introduces events and probability review. Chapter Two devotes to random variables in their two types: discrete and continuous with definitions of probability mass function, probability density function and cumulative distribution function as well. Chapter Three discusses mathematical expectation with its special types such as: moments, moment generating function and other related topics. Chapter Four deals with some special discrete distributions: (Discrete Uniform, Bernoulli, Binomial, Poisson, Geometric, Neg

      Faintly continuous (FC) functions, entitled faintly S-continuous and faintly δS-continuous functions have been introduced and investigated via a -open and -open sets. Several characterizations and properties of faintly S-continuous and faintly -Continuous functions were obtained. In addition, relationships between faintly s- Continuous and faintly S-continuous function and other forms of FC function were investigated. Also, it is shown that every faintly  S-continuous is weakly S-continuous. The Convers is shown to be satisfied only if the co-domain of the function is almost regular.

In this work we define and study new concept of fibrewise topological spaces, namely fibrewise soft topological spaces, Also, we introduce the concepts of fibrewise closed soft topological spaces, fibrewise open soft topological spaces, fibrewise soft near compact spaces and fibrewise locally soft near compact spaces.

In this paper we show that if ? Xi is monotonically T2-space then each Xi is monotonically T2-space, too. Moreover, we show that if ? Xi is monotonically normal space then each Xi is monotonically normal space, too. Among these results we give a new proof to show that the monotonically T2-space property and monotonically normal space property are hereditary property and topologically property and give an example of T2-space but not monotonically T2-space.

The main objectives of this pepper are to introduce new classes. We have attempted to obtain coefficient estimates, radius of convexity, Distortion and  Growth theorem and other related results for the classes

            This work, introduces some concepts in bitopological spaces, which are nm-j-ω-converges to a subset, nm-j-ω-directed toward a set, nm-j-ω-closed mappings, nm-j-ω-rigid set, and nm-j-ω-continuous mappings. The mainline idea in this paper is nm-j-ω-perfect mappings in bitopological spaces such that n = 1,2  and m =1,2 n ≠ m. Characterizations concerning these concepts and several theorems are studied, where j = q , δ, a , pre, b, b.

There are many methods of searching large amount of data to find one particular piece of information. Such as find name of person in record of mobile. Certain methods of organizing data make the search process more efficient the objective of these methods is to find the element with least cost (least time). Binary search algorithm is faster than sequential and other commonly used search algorithms. This research develops binary search algorithm by using new structure called Triple, structure in this structure data are represented as triple. It consists of three locations (1-Top, 2-Left, and 3-Right) Binary search algorithm divide the search interval in half, this process makes the maximum number of comparisons (Average case com

In this paper, we presented new types of Mc-function by using 𝜔-open and 𝑁-open sets some of them are weaker than Mc-function and some are stronger, which are 𝜔Mc-function, M𝜔c-function, 𝜔M𝜔c-function, 𝑁Mc-function, M𝑁c-function and 𝑁M𝑁c-function, also we submitted new kinds of continuous functions and compact functions and we illustrated the relationships between these types. The purpose of this paper is to expand the study of Mcfunction and to get results that we need to find the relationship with the types that have been introduced.

We introduce some new generalizations of some definitions which are, supra closure converge to a point, supra closure directed toward a set, almost supra converges to a set, almost supra cluster point, a set supra H-closed relative, supra closure continuous functions, supra weakly continuous functions, supra compact functions, supra rigid a set, almost supra closed functions and supra perfect functions. And we state and prove several results concerning it

This paper is concerned with introducing and studying the first new approximation operators using mixed degree system and second new approximation operators using mixed degree system which are the core concept in this paper. In addition, the approximations of graphs using the operators first lower and first upper are accurate then the approximations obtained by using the operators second lower and second upper sincefirst accuracy less then second accuracy. For this reason, we study in detail the properties of second lower and second upper in this paper. Furthermore, we summarize the results for the properties of approximation operators second lower and second upper when the graph G is arbitrary, serial 1, serial 2, reflexive, symmetric, tra