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, Negative Binomial and Hypergeometric) with their mathematical formulas of p.m.f., C.D.F. and m.g.f. Chapter Five deals with some special continuous distributions: (Uniform, Normal, Exponential, Gamma and Beta) with their mathematical formulas of p.m.f., C.D.F. and m.g.f. Many solved examples are intended in this book (obtaining mean and variance of distributions by m.g.f.). Chapter Six introduces univariate discrete and continuous transformations, i.e., one dimensional variables and their yielding probability distributions. Chapter Seven devotes to truncation of distributions from left, right or both sides, beside the probability distribution of order statistics. Chapter Eight discusses mathematical features of joint, marginal and conditional distributions, as well as independency via covariance and correlation of bivariate distributions. Chapter Nine deals with some special topics such as getting distribution for some transformation from multidimensional random variables by using moment generating function (m.g.f.) and cumulative distribution function (C.D.F.) Many solved examples (about 100) are intended in this book, in addition to a variety of unsolved relied problems (about 150) at the end of each chapter to enrich the statistical knowledge of our readers.
Mathematical Statistics - Second Edition
Mathematical Expectation
Moment Generating Function
Cumulative Distribution Function
Probability Mass Function
Probability Density Function
Discrete Distributions
Continuous Distributions
Joint distributions
Marginal Distributions
Conditional Distributions
Transformations
Order Statistics.
Publication Date
Sat Sep 01 2018
Journal Name
Polyhedron
Novel dichloro (bis {2-[1-(4-methylphenyl)-1H-1, 2, 3-triazol-4-yl-κN3] pyridine-κN}) metal (II) coordination compounds of seven transition metals (Mn, Fe, Co, Ni, Cu, Zn and Cd)
The 1
3-dipolar cycloaddition “click” reaction between an azide and an alkyne to give a 1
2
3-triazole was reported by Huisgen in 1961 [1]. In 2002 the Copper(I)-catalyzed Azide-Alkyne Cycloaddition (CuAAC) to prepare a 1
2
3-triazole was reported [2]
[3]. The existence of relatively basic nitrogen atoms in the 1
2
3-triazole rings
and the possibility of introducing additional donor groups in the substituents (Fig. 1)
made the CuAAC “click” reaction an attractive method to prepare differently substituted 1
2
3-triazoles. These compounds have been used as ligands to coordinate to various metal ions that display a range of applications such as in electrochemical and photochemical studies
in supramolecular chemistry
magnetism
metal-ion sensing and catalysis [4]. The reasons for the success of the “click” reaction
is that it is easy to carry out and is widely applicable. It is not affected by a variety of functional groups
and can be carried out with a variety of Cu(I) catalysts and solvents
including aqueous conditions. The Cu(I) catalysts overcome the high activation energy barrier of the non-catalyzed Huisgen reaction by changing the mechanism of the reaction. A large variety of copper catalysts can be used for the CuAAC reaction
on condition that the maximum concentration of Cu(I) species is generated during the reaction. The pre-catalyst can be a Cu(II) salt (usually CuSO4) together with a reducing agent (often sodium ascorbate) or a Cu(I) compound in the presence of a base or amine ligand and a reducing agent to prevent oxidation to Cu(II). Some strong oxidising cupric salts or complexes such as Cu(OAc)2 also work. The solvent is very flexible from organic to aqueous
with the most commonly used combination water + an alcohol (t-BuOH
MeOH or EtOH). The key role of the solvent or solvent mixture is to solubilize the substrates and Cu(I) catalyst in order to ensure rapid reactions. Such aqueous conditions are very useful for biochemical conjugations
as well as for organic syntheses.
We recently reported on the synthesis of a series of differently substituted 1
2
3-triazole chromophores
the substituted 2-(1-phenyl-1H-1
2
3-triazol-4-yl)pyridine ligands [5]
see Fig. 2 middle
with substituents R = H (L1)
CH3 (L2)
OCH3 (L3)
COOH
F
Cl
CN
CF3
O(CH2)3CH3 and N(CH3)2. These versatile ligands were found to coordinate to various first row transition metals
such as manganese
cobalt and nickel [6]. Here we extend the series to include more first row transition metal(II) coordination compounds
iron
copper and zinc
as well as a second row transition metal(II) coordination compound
cadmium
containing the 2-(1-(4-methyl-phenyl)-1H-1
2
3-triazol-1-yl)pyridine chromophore (Fig. 2 right with R = CH3). This series of seven novel coordination compounds is the first series of pyridyl-triazole based transition metal coordination compounds where seven different transition metals are coordinated to the same 1
2
3-triazole chromophore
namely 2-(1-(4-methyl-phenyl)-1H-1
2
3-triazol-1-yl)pyridine.