Second Stage


Advanced Calculus

Units : 8
Lecturer : Azhaar Hassan Sallo

Description : Course overview:

This course studies real sequences and infinite series without assumption of previous knowledge. The notions of upper and lower limits are introduced and used sparingly as a simplifying device; with their aid, the theory is given in almost complete form. The usual tests are given: in particular, the root test. With its aid, the treatment of power series is greatly simplified. Uniform convergence is presented with great care and applied to power series. Final sections point out the parallel with improper integrals; in particular, power series are shown to correspond to the Laplace transform. Also we opens with a review of determinants, and linear equations, and then develops matrix algebra, including Gaussian elimination, and n-dimensional geometry, with stress on linear We discuss partial derivatives and develops them with the aid of vectors (gradient, for example) and matrices; partial derivatives are applied to geometry and to maximum-minimum problems. The course introduces divergence and curl and the basic identities; orthogonal coordinates are treated concisely. One of the chapters is on integration, reviews definite and indefinite integrals, using numerical methods to show how the latter can be constructed; multiple integrals are treated carefully, with emphasis on the rule for change of variables; Leibnitz's Rule for differentiating under the integral sign is proved. Improper integrals are also covered; the discussion of these is completed , where they are related to infinite series. The concept of line and surface integrals is introduced. Although the notions are first presented without vectors, it very soon becomes clear how natural the vector approach is for this subject. Line integrals are used to provide an exceptionally complete treatment of transformation of variables in a double integral. Many physical applications. including potential theory, are given.

Course objective:

The goals of the Advanced Calculus course are to give an introduction to the subject of Real Analysis, and to continue the rigorous development of the theorems of Calculus to the areas of derivatives, integrals, and series of functions. Students should gain a more in depth understanding of these areas of Calculus and should increase their proficiency in proving various results. I hope to convey the usefulness of series of functions, the importance of the uniform norm, and the limitations of the Riemann integral. A focus is on learning to prove mathematical statements rigorously.

The Topics:

1.Infinite Sequences and Series.
1.1: Sequences. definition infinite sequence, convergence and divergence, definition diverges to infinity, calculating limits of sequences, using hopital's rule, bounded non decreasing sequences, decreasing sequence, bounded, upper bound, least upper bound
1.2: Infinite Series, Infinite Series, Geometric Series, Divergent Series, The nth-Term Test for Divergence, Combining Series, The Integral Test, The p –Series, The Comparison Test, The Limit Comparison Test, The Ratio and Root Tests.
1.3: Alternating Series. Absolute and Conditional Convergence. The Absolute Convergence Test.
2.4: Power Series. Power Series and Convergence. The Radius of Convergence of a Power Series. How to Test a Power Series for Convergence
2.5: Taylor and Maclaurin Series.

2. Polar Coordinates
2.1 : Definition of Polar Coordinates.
2.2 : Polar Equations and Graphs.
2.3 : Relating Polar and Cartesian Coordinates.
2.4 : Graphing in Polar Coordinates. Symmetry. Symmetry Tests for Polar Graphs. Finding Points Where Polar Graphs Intersect.
2.5 : Areas in Polar Coordinates.
2.6 : Length of a Polar Curve.

3. Vectors and the Geometry of Space.
3.1: Vectors. Vector Addition and Multiplication of a Vector by a Scalar. Properties of Vector Operations. Unit Vectors.
3.2: The Dot Product. Angle Between Vectors. Perpendicular (Orthogonal) Vectors. Properties of the Dot Product
3.3: The Cross Product. The Cross Product of Two Vectors in Space. Parallel Vectors. Properties of the Cross Product.
3.4: Lines and Planes in Space. Vector Equation for a Line. Parametric Equations for a Line. The Distance from a Point to a Line in Space. An Equation for a Plane in Space.Equation for a Plane.Lines of Intersection. The Distance from a Point to a Plane. Angles Between Planes

4. Partial Derivatives.
4.1: Functions of Several Variables. Domains and Ranges.
4.2: Limits and Continuity in Higher Dimensions. Limits. Limit of a Function of Two Variables. Properties of Limits of Functions of Two Variables. Continuity. Continuous Function of Two Variables.
4.3: Partial Derivatives. Partial Derivatives of a Function of Two Variables. Functions of More Than Two Variables. Partial Derivatives and Continuity. The Chain Rule.
4.4: Directional Derivatives and Gradient Vectors. Directional Derivatives in the Plane. Gradient Vector.
4.5: Tangent Planes and Differentials.
4.6: Extreme Values and Saddle Points. Lagrange Multipliers.

5. Multiple Integrals.
5.1: Double Integrals. Double Integrals over Rectangles. Double Integrals as Volumes. Fubini’s Theorem for Calculating Double Integrals. Properties of Double Integrals. Areas of Bounded Regions in the Plane. Double Integrals in Polar Form.
5.2: Integrals in Polar Coordinates. Area in Polar Coordinates. Changing Cartesian Integrals into Polar Integrals.
5.3:Triple Integrals in Rectangular Coordinates. Triple Integrals. Volume of a Region in Space. Properties of Triple Integrals.
5.4: Triple Integrals in Cylindrical and Spherical Coordinates. Integration in Cylindrical Coordinates. Equations Relating Rectangular (x,y,z) and Cylindrical Coordinates. Spherical Coordinates and Integration. Equations Relating Spherical Coordinates to Cartesian and Cylindrical Coordinates.
5.5: Substitutions in Multiple Integrals.

6. Integration in Vector Fields.
6.1: Line Integrals, Vector Fields, Gradient Fields
6.2: Green's Theorem in the Plane. Using Green’s Theorem to Evaluate Line Integrals.
6.3: Surface Area and Surface Integrals. Surface Area. Surface Integrals. Parametrized Surfaces.
6.4:Stokes' Theorem.
6.5: The Divergence Theorem.

Course Reading List and References :
- Advanced Calculus 3rd Edition by Taylor Angus & Wiley. Fayez, 1955.
- Calculus, Haward Anton, Eighth Edition, 2005.
- Schaum's Outline Theory and Problems in Advanced Calculus Robert Wrede, Murray R. Spiegel. 2002 by The McGraw -Hill Companies.
- Thomas’ Calculus, 2005 Pearson Education, Inc., publishing as Pearson Addison – Wesley.
-Calculus, Haward Anton, Eighth Edition, 2005

Classical Mechanics

Units : 6
Lecturer : Dr. Haval Younis Yacoob

Description :

Course overview:

This course ordinarily forms the first part of the college sequence that serves as the foundation in physics for students majoring in the physical, Mathematical sciences and engineering. The sequence is parallel to or preceded by mathematics courses that include calculus. Methods of calculus are used wherever appropriate in formulating physical principles and in applying them to physical problems. Strong emphasis is placed on solving a variety of challenging problems, some requiring calculus. The subject matter of the Mechanics course is classical mechanics and includes topics in kinematics; Newton’s laws of motion, work, energy and power; systems of particles and linear momentum; circular motion and rotation; oscillations; and gravitation. Mechanics course is the first part of a sequence which in college is a very intensive one-year course. Use of calculus in problem solving and in derivations is expected to increase as the course progresses. Calculus is used freely in formulating principles and in solving problems. Please note: Although fewer topics are covered in Mechanics, but they are covered in greater depth and with greater analytical and mathematical sophistication, including calculus applications. This course is designed to provide students with a working knowledge of the elementary physics principles mentioned above, as well as their applications, and to enhance their conceptual understanding of physical laws.

Course objective:
1. How to use Newton’s laws of motion to solve advanced problems involving the dynamic motion of classical mechanical systems.
2. How to use calculus and other advanced mathematics in the solution of the problems considered in the above topics.
3. How to use conservation of energy and linear and angular momentum to solve dynamics problems.
4. How to represent the equations of motion for complicated mechanical systems using the Newtonian formulations of classical mechanics.
5. The intended goal of this course is the development of an ability to think in a critical manner using both concrete and abstract examples from physics as models, and
6. Apply calculus techniques in solving word problems

Student learning outcome:
The general outcomes of this course are:

- Students should be able to demonstrate their understanding of the foundations in mechanics by demonstrating competence in the major through appropriate homework assignments and examinations.

- Students should be able to demonstrate competency in experimental design and scientific data collection and analysis.

- Students will be able to competently solve appropriate problems using increasingly important computational and mathematical tools.

- Students should be able to apply calculus techniques in solving word problems

- Students should be able to demonstrate an understanding of the impact of mechanics issues on society.


Chapter (1) Kinematics
1- Motion in 1-D
2-Motion in 2-D
a- Projectiles
b- Uniform Circular Motion

Chapter (2)Dynamics of particle
1-The law of inertia
2-The concept of mass.
3-Linear momentum.
4- Principle of conservation of momentum.
5- Newton's second law. Newton's third law.
6- The concept of force.
7- Central forces
Chapter (3) Work and Energy
1-Work & Power.
2- The work, energy theorem
3- Kinetic energy.
4- Work of a constant force.
5- Potential energy.
6- Work done by a central force, potential energy. Conservative forces,
7- Conservative & Neoconservative forces.

Chapter (4) Dynamics of the a system of particles
1- Center of mass
2-Reduce mass
3- Angular momentum & kinetic energy of a system of particles.
4- Collision: elastic, inelastic

Chapter (5) Dynamics of Rigid body
1-Angular momentum of rigid body.
2-Moment of inertia.
3-Calculation of the moment of inertia.
4- Equation of motion for rotation of rigid body.
5- Kinetic energy of rotation

Chapter (6) Oscillatory Motion
1-Harmonic motion
2-Kinematics of simple harmonic motion.
3-Dynamics of SHM. Simple pendulum.

Course Reading List and References :
-Fundamentals of Physics by David Halliday, Robert Resnick, and Jearl Walker
- Physics by Cutnell and Johnson 7th Edition 2007

Computer Application - Matlab

Units : 6
Lecturer : Huda Younus Najm

Description : Course overview:
MATLAB (matrix laboratory) is a high-performance language for technical computing. It integrates
computation, visualization, and programming in an easy-to-use environment where problems and
solutions are expressed in familiar mathematical notation. Typical uses include: Math and computation , Algorithm development, Modeling, simulation, and prototyping, Data analysis, exploration, and visualization, Scientific and engineering graphics, Application development, including Graphical User Interface building . MATLAB is a programming language developed by MathWorks. It started out as a matrix programming language where linear algebra programming was simple. It can be run both under interactive sessions and as a batch job. This course gives you aggressively a gentle introduction of MATLAB programming language. It is designed to give students fluency inMATLAB programming language. Problem-based MATLAB examples have been given in simple and easy way to make your learning fast and effective. This course has been prepared for the beginners to help them understand basic to advanced functionality of MATLAB. It allows matrix manipulations; plotting of functions and data; implementation of algorithms; creation of user interfaces; interfacing with programs written in other languages, including C, C++, Java, and FORTRAN; analyze data; develop algorithms; and create models and applications. It has numerous built-in commands and math functions that help you in mathematical calculations, generating plots, and performing numerical methods.

Course objective:
To develop computing skills and to reinforce mathematical concepts by computational example through the medium of MATLAB, a general purpose, high-level, numerical and graphical computing package. MATLAB is used in every facet of computational mathematics. Following are some commonly used mathematical calculations where it is used most commonly:
- Dealing with Matrices and Arrays
- 2-D and 3-D Plotting and graphics
- Linear Algebra
- Algebraic Equations
- Non-linear Functions
- Statistics
- Data Analysis
- Calculus and Differential Equations
- Numerical Calculations
- Integration
- Transforms
- Curve Fitting
- Various other special functions

Student learning outcome:
A student who completes this module successfully should be able to:
MATLAB is widely used as a computational tool in science and engineering encompassing the fields of physics, chemistry, math and all engineering streams. It is used in a range of applications including:
- signal processing and Communications
- image and video Processing
- control systems
- test and measurement
- computational finance
- computational biology
- Study Master, PhD because students will be obtained what is needed for them.
After completing this tutorial you will find yourself at a moderate level of expertise in using MATLAB from where you can take yourself to next levels. Matlab is interactive environment for numerical computation, visualization and programming.

The Topics:
Section 1: Starting with MATLAB
Section 2 : Creating Arrays
Section 3 : Mathematical Operations with Arrays
Section 4 : Script Files
Section 5 : Programming in MATLAB
Section 6 : Two-Dimensional Plots
Section 7 : Functions and Function Files
Section 8 : Polynomials, Curve Fitting, and Interpolation
Section 9 : Three-Dimensional Plots
Section 10: solving linear system, limit, diff., integration
Section 11: introduction to image Processing

Course Reading List and References :
- Rao V.Dukkipati , "Matlab qn introduction with applications ", New age international , (2014).
- Ela Pekalska , Marjolein van der Glas, "Introduction to MATLAB " , Pattern Recognition
Group, Faculty of Applied Sciences, Delft University of Technology (2002).
- Internet website
- Anything you want you can Google it.

Linear Algebra

Units : 6
Lecturer : Nechirvan Badal Ibrahim

Description : Course overview:

This course book is designed to teach the university mathematics student the basics of the subject of linear algebra. There are no prerequisites other than ordinary algebra, but it is probably best used by a student who has the 'mathematical maturity' of a sophomore or junior. The text has two goals: to teach the fundamental concepts and techniques of matrix algebra and abstract vector spaces and to teach the techniques associated with understanding the definitions and theorems forming a coherent area of mathematics. So there is an emphasis on worked examples of nontrivial size and on proving theorems carefully. The first half of this text is basically a course in matrix algebra, though the foundation of some more advanced ideas is also being formed in these early sections. Vectors are presented exclusively as column vectors, and linear combinations are presented very early. Spans, null spaces and column spaces are also presented early, simply as sets, saving most of their vector space properties for later, so they are familiar objects before being scrutinized carefully.

Course objective:
This course aims at providing students with the essential concepts of linear algebra. The concepts of linear algebra are discussed using skills and concepts acquired in the fundamental subjects in mathematics. The course serves to illustrate how mathematics concepts can be applied to explain the various properties of linear algebra. Topics covered include: systems of linear equations and matrices, determinants and it is applications, vectors and vector space, eigenvalues and eigenvectors, linear transformations, linear mappings and matrices, inner product, spaces, orthogonality, canonical forms, linear functional and the dual space .

Student learning outcome:
After completing Linear Algebra, the student should be able to:
- Prove elementary statements concerning the theory of systems of linear equations.
- Solve application problems of systems of linear equations.
- Perform the operations of addition, scalar multiplication, multiplication, and find the inverses and transposes of matrices.
- Calculate determinants using row operations, column operations, and expansion down any column or across any row.
- Prove elementary statements concerning the theory of matrices and determinants.
- Prove algebraic statements about vector addition, scalar multiplication, inner products, projections, norms, orthogonal vectors, linear independence, spanning sets, subspaces, bases, dimension and rank.
- Write the relationships between A (being invertible) , det A, AX = 0 having a solution, the rank of A and linear independence.
- Find the kernel, rank, range and nullity of a linear transformation.
- Calculate eigenvalues, eigenvectors and eigen spaces.
- Solve projected steady state dynamical systems.
- Determine if a matrix is diagonalizable, and if it is, diagonalize it.

The Topics:

1.2 Matrices
1.3 Matrix Addition and Scalar Multiplication
1.4 Summation Symbol
1.5 Matrix Multiplication
1.6 Transpose of a Matrix
1.7 Square Matrices
1.8 Powers of Matrices, Polynomials in Matrices
1.9 Invertible (Nonsingular) Matrices
1.10 Special Types of Square Matrices
1.11 Complex Matrices
1.12 Block Matrices.

2.1 Introduction
2.2 Determinants of Orders 1 and 2
2.3 Determinants of Order 3
2.4 Permutations
2.5 Determinants of Arbitrary Order
2.6 Properties of Determinants
2.7 Minors and Cofactors
2.8 Evaluation of Determinants
2.9 Classical Adjoint
2.10 Submatrices,
2.11 Block Matrices and Determinants
2.12 Determinants and Volume.


3.1 Introduction
3.2 Polynomials of Matrices
3.3 Characteristic Polynomial, Cayley–Hamilton Theorem
3.4 Diagonalization, Eigenvalues and Eigenvectors
3.5 Computing Eigenvalues and Eigenvectors, Diagonalizing Matrices
3.6 Diagonalizing Real Symmetric Matrices and Quadratic Forms.

4.1 Introduction
4.2 Basic Definitions, Solutions
4.3 Equivalent Systems, Elementary Operations
4.4 Small Square Systems of Linear Equations
4.5 Systems in Triangular and Echelon Forms
4.6 matrix inverse method
4.7 Cramer’s Rule
4.8 Gaussian Elimination
4.9 Echelon Matrices, Row Canonical Form, Row Equivalence
4.10 Matrix Equation of a System of Linear Equations
4.11 Homogeneous Systems of Linear Equations.

5.1 Introduction
5.2 Vectors in Rn
5.3 Vector Addition and Scalar Multiplication
5.4 Dot (Inner) Product
5.5 Located Vectors, Hyperplanes, Lines, Curves in Rn
5.6 Vectors in R3 (Spatial Vectors), ijk Notation
5.7 Complex Numbers
5.8 Vectors in Cn
5.9 Vector Spaces
5.10 Examples of Vector Spaces
5.11 Linear Combinations, Spanning Sets
5.12 Subspaces
5.13 Linear Spans, Row Space of a Matrix
5.14 Linear Dependence and Independence
5.15 Basis and Dimension
5.16 Application to Matrices, Rank of a Matrix
5.17 Sums and Direct Sums
5.18 Coordinates.

6.1 Introduction
6.2 Mappings, Functions
6.3 Linear Mappings (Linear Transformations)
6.4 Kernel and Image of a Linear Mapping
6.5 Singular and Nonsingular Linear Mappings, Isomorphisms
6.6 Operations with Linear Mappings
6.7 Algebra A(V ) of Linear Operators.


7.1 Introduction
7.2 Matrix Representation of a Linear Operator
7.3 Change of Basis
7.4 Similarity
7.5 Matrices and General Linear Mappings.

8.1 Introduction
8.2 Inner Product Spaces
8.3 Examples of Inner Product Spaces
8.4 Cauchy–Schwarz Inequality, Applications
8.5 Orthogonality
8.6 Orthogonal Sets and Bases
8.7 Gram–Schmidt Orthogonalization Process
8.8 Orthogonal and Positive Definite Matrices
8.9 Complex Inner Product Spaces
8.10 Normed Vector Spaces (Optional).


9.1 Introduction
9.2 Triangular Form
9.3 Invariance
9.4 Invariant Direct-Sum Decompositions
9.5 Primary Decomposition
9.6 Nilpotent Operators
9.7 Jordan Canonical Form
9.8 Cyclic Subspaces
9.9 Rational Canonical Form
9.10 Quotient Spaces.

10.1 Introduction
10.2 Linear Functionals and the Dual Space
10.3 Dual Basis
10.4 Second Dual Space
10.5 Annihilators
10.6 Transpose of a Linear Mapping.

Course Reading List and References :
1. Bernard Kolman Third Elman”Introductory Linear Algebra with Applications”.
2. Howard Anton Chris Rorres “Elementary Linear Algebra”.
3. James B. Carrell “Fundamentals of Linear Algebra’’ (July, 2005).
4. Jim Hefferon, Linear Algebra, 2014.
5. Richard O.Hill, Jr." Elementary Linear Algebra”.
6. Seymour Lipschutz and Marc Lars Lipson, Linear Algebra Schaum’s Outline Series, 4th Edition ( 2009).
7. David, C. Lay, Linear Algebra and Its Applications , 4th Edition,(2012).
8. Tom D. and Andrew W., Linear Algebra in Twenty Five Lectures. March 27, 2012.

Ordinary Differential Equations

Units : 6
Lecturer : Ahmed Muhammed Hussien

Description : Course overview:

The aim of this module is to introduce the students to the basic theory of ordinary differential equations and give a competence in solving ordinary differential equations by using different methods of solution of differential equations.

Course objective:
The subject of differential equations is a very important branch of applied mathematics. Many phenomena from physics, biology and engineering may be described using ordinary differential equations. They are also used to model the behaviour of systems in the natural world, and predict how these systems will behave in the further. For instance, exponential growth (the rate of change of a population is proportional to the size of the population) is expressed by the differential equation . Newton's Law of Gravitation (acceleration is inversely proportional to the square of distance) translates to the equation . Many examples are found in the fields of physics, engineering, biology, chemistry and economics.
The traditional course in differential equations focused on the small number of differential equations for which exact solutions exist. How-ever, the methods used by scientists today have changed dramatically due to computer (using different type of computational package like Maple, Mathematica, reduce, Singular, etc). Here we will cover almost all methods for solving every kind of ordinary differential equations.

Student learning outcome:
A student who completes this module successfully should be able to:
- The student will learn to formulate ordinary differential equations (ODEs) and seek
understanding of their solutions.
- The student will recognise basic types of differential equations which are solvable,
and will understand the features of linear equations in particular.
- Students will be familiar to derive methods to solve ordinary differential equations.
- Study Master, PhD because students will be obtained what is needed for them.

The Topics:
Chapter One: Definitions and elimination of essential constants
1.1 Definitions and examples of differential equations.

1.2 Some observation concern solutions of differential
1.3 The elimination of essential arbitrary constants.
1.4 Geometrical interpretation of differential equations.

Chapter Two: Equations of first order and first degree
2.1 Separable differential equations.
2.2 Homogeneous differential equations.
2.3 Coefficients linear in the two variables.
2.4 Exact differential equations.
2.5 Integrating factors.
2.6 First order linear differential equations.
2.7 Equations reducible to liner form.
2.7.1 Bernoulli equations.
2.7.2 Riccati equations.
2.8 substitution suggested by the equation.
2.9 Simultaneous first order differential equations.

Chapter Three: The equation is of first order and of second or higher degree
3.1 Factoring the lift member (equations that can be
factorized into factors of first degree).
3.2 Equations solvable for y.
3.3 Equations solvable for x.

Chapter Four: Linear differential equations with constant coefficients

4.1 The general linear equations.
4.2 Linear independence.
4.3 The wronskian.
4.4 Differential operator D.
4.5 Homogeneous linear differential equations with constant coefficients.
4.6 Properties of operator D.
4.7 Non-homogeneous linear differential equations with constant coefficients.
4.8 Reduction of order.
4.9 Variation of parameters.

Chapter Five: Differential equations with variable coefficients
5.1 Dependent variable missing.
5.2 Independent variable missing.
5.3 The Chaucy and Legender differential equations.
5.4 Second order linear differential equations with variable coefficients.
5.5 Finding the particular solution of a homogeneous linear differential equations with variable coefficients.

Chapter Six: The Laplace transformation (Laplace's transform and its application to differential  equations)
6.1 Definition of the Laplace transform.
6.2 Laplace transform of some elementary functions.
6.3 Certain theorems for Laplace's transform.
6.4 Laplace transform of a derivative.
6.5 Inverse Laplace transform.
6.6 Properties of inverse Laplace transform.
6.7 transforming initial value problems.
6.8 Derivative of the Laplace transform.

Chapter Seven: The power series method
7.1 Ordinary points and singular points.
7.2 Solutions a round ordinary points.
7.3 Frobenius series solution.
7.3.1 Indicial equation with difference of roots noninteger.
7.3.2 Indicial equation with equal roots.
7.3.3 Indicial equation with difference of roots a positive integer, non-logarithmic case.
7.3.4 Indicial equation with difference of roots a positive integer, logarithmic case.

Chapter Eight: Systems of differential equations
8.1 Definitions and theorems regarding systems of differential equations.
8.2 Fundamental matrices.
8.3 Abel's identity.
8.4 Homogeneous systems.
8.5 Non-homogeneous systems.

Course Reading List and References :
1) Differential Equations: a modelling approach. By Frank R. Giordano and Maurice D. Weir.
2) Elementary Differential Equations. By Earl D. Rainvlle and Philip E. Bedient.
3) Elementary Differential Equations with Linear Algebra. By Ross L Finney and Donald R. Ostbery.
4) Ordinary Differential Equations. By Tyn Myint-V .
5) Differential Equations and Boundary Value Problems. By C. Henry Edward and David E. Penney.
6) Applied Differential Equations. By Murray R. Spoegel .
7) Differential Equations. By C. Ray Wylie.
8) Schaum's Outline Series, Theory and problems of Differential Equations. By Frank Ayres, JR. including 560 solved problems.
9) Schaum's: 2500 solved problem in Differential Equations. By Richard Bronson.

Probability and Statistics

Units : 6
Lecturer : Muhammad A. Sadiq

Description :

Course overview:

This class is an introduction to probability theory. Probability has direct relation with our daily life, for example if somebody wants to buy a car and has three options in probability we can measure each option for choosing. Also studding this module with respect to mathematicians is important because it is a field to study Mathematical Statistics and The concept we will cover in this module are counting techniques, probability theory, probability density function and mass function and joint distribution. First we study the strategy of counting technique then measure the probability via axioms and studding some operations on it, this is the first step to built probability density and mass functions. In the last century, scientists a plied probability to finance and they invented some new markets and contract, this has been done via black schole model which.

Course objective:
This subject is intended to:
The purpose of this module is to broaden the students' knowledge and experience of probability theory by studying a mathematically fully rigorous framework for the subject. This module is in the Probability Pathway. It builds upon the concepts of probability introduced in high school and in the first year and develops these concepts in a mathematically fully rigorous manner. Students will acquire knowledge and skills of measure-theoretic probability that are fundamental to many areas of current research in probability and statistical theory.

Student learning outcome:

At the end of this course and having completed the Essential reading and activities, you should be able to: Compute probabilities by modeling sample spaces and applying rules of permutations and combinations, additive and multiplicative laws and conditional probability a, b Construct the probability distribution of a random variable, based on a real-world situation and use it. To compute expectations and variances. Compute probabilities based on practical situations using some special distributions. Furthermore students should able to compute the expectation and conditional expectation as well as know how to compute jointly distribution function and marginal distribution.

The Topics:

- Basic statistical measure: Arithmetic mean, weighted arithmetic mean, median, variance and standard deviation.
- Probability: Sample space, events, axioms and counting techniques.
- Conditional probability: Independence and Bay’s Theorem and examples
- Discrete and continuous random variables and probability distributions: Probability distributions for both discrete and continuous random variables.
- Expectation: Expectation values of discrete and continuous random variables.
- Moments of Random Variables: Moments of Random Variables, variance of Random Variables and Moment Generating Functions
- Probability distributions: Discrete and continuous distributions.
- Joint distribution function: Jointly distributed random variables, expected values, covariance and correlation.

Course Reading List and References :
- Probability and Statistic. Jay L. Devore (Library)
- Probability and Mathematical Statistic. Prasana. S. (online)
- Introduction to Probability Theory and Statistics. David S. (online)