Introductory Mathematics and Statistics for Islamic Finance -- Contents -- Preface -- Acknowledgments -- About the Authors -- Part One: Mathematics -- Chapter 1: Elementary Mathematics -- BASIC MATHEMATICAL OBJECTS -- Real Numbers -- Complex Numbers -- Absolute Value of a Number -- Vectors and Arrays -- Angles and Directions -- Graphics -- Reporting Economic and Financial Data -- VARIABLES, MONOMIALS, BINOMIALS, AND POLYNOMIALS -- Monomials, Binomials, and Polynomials -- Polynomial Lags -- Identities -- Factorization of a Polynomial -- EQUATIONS -- EQUATIONS OF HIGHER ORDER -- SEQUENCES -- SERIES -- Convergence of a Series -- APPLICATIONS OF SERIES TO PRESENT VALUE OF ASSETS -- Applications of Series to Present Value Computation -- SUMMARY -- QUESTIONS -- Chapter 2: Functions and Models -- DEFINITION OF A FUNCTION -- Parametric Form of a Function -- FUNCTIONS AND MODELS IN ECONOMICS -- The Market Model: Demand and Supply Functions -- The Budget Constraint -- The Production Possibility Frontier (PPF) -- The Utility Function -- Production Function -- Other Functions in Economics -- FUNCTIONS AND MODELS IN FINANCE -- The Present Value Function -- The Capital Asset Pricing Model (CAPM) -- Payoff of a Futures Contract -- Payoff of an Option Contract -- Payoff to a Swap -- Price of an Option -- The Forward Exchange Rate -- MULTIVARIATE FUNCTIONS IN ECONOMICS AND FINANCE -- Parametric Representation -- Level Curves -- SUMMARY -- QUESTIONS -- Chapter 3: Differentiation and Integration of Functions -- DIFFERENTIATION -- DIFFERENTIATION RULES -- MAXIMUM AND MINIMUM OF A FUNCTION -- MEAN VALUE THEOREM -- POLYNOMIAL APPROXIMATIONS OF A FUNCTION: TAYLOR'S EXPANSION -- INTEGRATION -- Integration -- The First Fundamental Theorem of Calculus -- Second Fundamental Theorem of Calculus -- Change in Variables in Indefinite Integrals -- Double Integral.

APPLICATIONS IN FINANCE: DURATION AND CONVEXITY OF A SUKUK -- Duration of a Sukuk -- Application of Taylor Expansion to the Convexity of Sukuk's Price -- SUMMARY -- QUESTIONS -- Chapter 4: Partial Derivatives -- DEFINITION AND COMPUTATION OF PARTIAL DERIVATIVES -- The Chain Rule -- Derivatives of Implicit Functions -- TOTAL DIFFERENTIAL OF A FUNCTION WITH MANY VARIABLES -- DIRECTIONAL DERIVATIVES -- GRADIENTS -- TANGENT PLANES AND NORMAL LINES -- Tangent Planes -- Normal Line -- EXTREMA OF FUNCTIONS OF SEVERAL VARIABLES -- EXTREMAL PROBLEMS WITH CONSTRAINTS -- Elimination Method -- Lagrange Method -- SUMMARY -- QUESTIONS -- Chapter 5: Logarithm, Exponential, and Trigonometric Functions -- LOGARITHM FUNCTIONS -- Logarithm Identities -- Change of Base -- The Natural Logarithmic Function -- THE EXPONENTIAL FUNCTION -- POWER SERIES OF LOGARITHMIC AND EXPONENTIAL FUNCTIONS -- GENERAL EXPONENTIAL AND LOGARITHMIC FUNCTIONS -- SOME APPLICATIONS OF LOGARITHM AND EXPONENTIAL FUNCTIONS IN FINANCE -- Simple Compounding and Continuous Compounding of Returns -- The Present Value Formula -- The Normal Distribution -- INTEGRATION BY PARTS -- TRIGONOMETRIC FUNCTIONS -- SUMMARY -- QUESTIONS -- Chapter 6: Linear Algebra -- VECTORS -- Addition of Vectors -- Multiplication of Vectors -- Vector Space -- Linear Combinations of Vectors -- Linear Dependence and Linear Independence of Vectors -- Bases of a Vector Space -- MATRICES -- Transposes of Matrices -- Matrix Multiplication -- SQUARE MATRICES -- Symmetric Matrix -- Positive Definite Matrix -- Quadratic Forms -- Orthogonal Matrix -- THE RANK OF A MATRIX -- DETERMINANT OF A SQUARE MATRIX -- HOMOGENOUS SYSTEMS OF EQUATIONS -- INVERSE AND GENERALIZED INVERSE MATRICES -- Generalized Inverse of a Matrix -- EIGENVALUES AND EIGENVECTORS -- Similarity of Square Matrices -- Diagonable Matrix -- Cholesky Decomposition.

STABILITY OF A LINEAR SYSTEM -- APPLICATIONS IN ECONOMETRICS -- SUMMARY -- QUESTIONS -- Chapter 7: Differential Equations -- EXAMPLES OF DIFFERENTIAL EQUATIONS -- SOLUTION METHODS FOR THE DIFFERENTIAL EQUATION -- Method of Indefinite Integrals -- Method of Separable Variables -- FIRST-ORDER LINEAR DIFFERENTIAL EQUATIONS -- SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS -- Homogeneous Linear Differential Equation -- Nonhomogeneous Linear Differential Equations -- LINEAR DIFFERENTIAL EQUATION SYSTEMS -- Transforming the System into a Second-Order Homogeneous Differential Equation -- Method of Eigenvalues and Eigenvectors -- PHASE DIAGRAMS AND STABILITY ANALYSIS -- Phase Line of an Ordinary Differential Equation -- Phase Diagram of a Linear Differential Equation System -- SUMMARY -- QUESTIONS -- Chapter 8: Difference Equations -- DEFINITION OF A DIFFERENCE EQUATION -- FIRST-ORDER LINEAR DIFFERENCE EQUATIONS -- Solutions of the First-Order Difference Equation -- The Impulse Response Function -- The Cobweb Model -- SECOND-ORDER LINEAR DIFFERENCE EQUATIONS -- Homogeneous Second-Order Difference Equations -- Nonhomogeneous Second-Order Difference Equations -- The Multiplier-Accelerator Model -- SYSTEM OF LINEAR DIFFERENCE EQUATIONS -- EQUILIBRIUM AND STABILITY -- Conditions for Stability -- Stability of the Linear Difference System -- Phase Plane -- SUMMARY -- QUESTIONS -- Chapter 9: Optimization Theory -- THE MATHEMATICAL PROGRAMMING PROBLEM -- Formulation of the Programming Problem -- The Geometry of Optimization -- UNCONSTRAINED OPTIMIZATION -- One Variable Function y= F(x) -- Function of Two Variables z= F(x,y) -- Definitions -- CONSTRAINED OPTIMIZATION -- The Method of Lagrange Multipliers -- THE GENERAL CLASSICAL PROGRAM -- The Geometry of Constrained Optimization -- Interpretation of the Lagrangian Multipliers -- Nonlinear Programming.

The Case of No Inequality Constraints -- The Kuhn-Tucker (K-T) Conditions -- SUMMARY -- QUESTIONS -- Chapter 10: Linear Programming -- FORMULATION OF THE LP -- Standard Form and Canonical Form of the LP -- The Geometry of the LP -- THE ANALYTICAL APPROACH TO SOLVING AN LP: THE SIMPLEX METHOD -- Notion of Technical Equivalence -- The Simplex Method -- THE DUAL PROBLEM OF THE LP -- THE LAGRANGIAN APPROACH: EXISTENCE, DUALITY, AND COMPLEMENTARY SLACKNESS THEOREMS -- Interpretation of the Dual Variables -- ECONOMIC THEORY AND DUALITY -- SUMMARY -- QUESTIONS -- Part Two: Statistics -- Chapter 11: Introduction to Probability Theory: Axioms and Distributions -- THE EMPIRICAL BACKGROUND: THE SAMPLE SPACE AND EVENTS -- Experiment -- Sample Space -- Events -- DEFINITION OF PROBABILITY -- Axioms of Probability -- RANDOM VARIABLE -- TECHNIQUES OF COUNTING: COMBINATORIAL ANALYSIS -- Factorial Notation -- Permutations -- Ordered Samples -- Combinations -- Tree Diagrams -- CONDITIONAL PROBABILITY AND INDEPENDENCE -- Conditional Probability -- Bayes' Theorem -- Independence of Events -- PROBABILITY DISTRIBUTION OF A FINITE RANDOM VARIABLE -- Probability Distribution and Histogram -- Cumulative Distribution Function -- Continuous Random Variables -- MOMENTS OF A PROBABILITY DISTRIBUTION -- First Moment of the Random Variable -- Second Moment of the Random Variable: Variance and Standard Deviation -- Third Moment of a Random Variable: Skewness -- Fourth Moment of a Random Variable: Kurtosis -- JOINT DISTRIBUTION OF RANDOM VARIABLES -- Independent Random Variables -- CHEBYSHEV'S INEQUALITY AND THE LAW OF LARGE NUMBERS -- Chebyshev's Inequality -- Law of Large Numbers -- The Central Limit Theorem -- SUMMARY -- QUESTIONS -- Chapter 12: Probability Distributions and Moment Generating Functions -- EXAMPLES OF PROBABILITY DISTRIBUTIONS -- The Uniform Distribution.

The Bernoulli Distribution -- The Binomial Distribution -- The Poisson Distribution -- The Normal Distribution -- The Chi-Square Distribution -- The t Distribution -- The F Distribution -- EMPIRICAL DISTRIBUTIONS -- MOMENT GENERATING FUNCTION (MGF) -- Examples of Moment Generating Functions -- SUMMARY -- QUESTIONS -- Chapter 13: Sampling and Hypothesis Testing Theory -- SAMPLING DISTRIBUTIONS -- Sampling Distribution of the Mean -- Sampling Distribution of Proportions -- Sampling Distribution of Differences -- ESTIMATION OF PARAMETERS -- Unbiased Estimates -- Efficient Estimates -- Point Estimates and Interval Estimates -- CONFIDENCE-INTERVAL ESTIMATES OF POPULATION PARAMETERS -- Confidence Intervals for Means -- Confidence Intervals for Proportions -- Confidence Intervals for Differences -- Confidence Intervals for Standard Deviations -- HYPOTHESIS TESTING -- Statistical Hypotheses -- Null Hypotheses -- Alternative Hypothesis -- Type I and Type II Errors -- Level of Significance -- Probability Value: p-Value -- Special Tests -- Means -- Proportions -- TESTS INVOLVING SAMPLE DIFFERENCES -- SMALL SAMPLING THEORY -- Tests Based on the Student's t-Distribution -- Confidence Intervals for t Distribution -- Tests Based on the Chi-Square Distribution -- Confidence Intervals for χ² -- Tests Based on the F Distribution -- SUMMARY -- QUESTIONS -- Chapter 14: Regression Analysis -- CURVE FITTING -- Equations of Approximating Curves -- The Linear Regression Line -- The Principle of Estimation: Method of Least Squares -- LINEAR REGRESSION ANALYSIS -- Formulation of the Regression Model -- Estimation of the Regression Model -- The Method of Moments -- The Method of the Maximum Likelihood -- Estimate of the Variance of the Error Term -- The Coefficient of Determination and the Coefficient of Correlation.

THE PROBABILITY DISTRIBUTION OF THE ESTIMATED REGRESSION COEFFICIENTS â AND b.