TABLE of CONTENTS
1 Probability Distribution of Functions Of Random Variables
1.1 Preliminaries
1.1.1 Probability Mass and Density Functions
1.1.2 The Cumulative Distribution Function (CDF)
1.1.3 The Moment Generating Function (MGF)
1.1.4 Multivariate Distributions and Independence
1.1.5 Joint Moment Generating Functions
1.1.6 Properties of the Bivariate CDF
1.2 Distribution of Order Statistics
1.2.1 The Joint PDF
1.2.2 Derivation of Extremes: Minimum and Maximum
1.2.3 The \(k^{\text {th}}\) Order Statistic
1.2.4 Distribution of the Range
1.3 Functions of Random Variables
1.3.1 The Cumulative Distribution Function Method
1.3.2 Transformation (Change of Variable) Method
1.3.3 Distributions of Sum and Difference of Random Variables
1.3.4 Distributions of Products and Quotients of Random Variables
1.3.5 Moment Generating Function Method
1.3.6 Distribution of \(t\),\(\chi ^2\) and \(F\) Random Variables
1.4 Convergence and Limit Theorems
1.4.1 Chebyshev’s Inequality
1.4.2 Convergence in Probability
1.4.3 The Weak Law of Large Numbers
1.4.4 Convergence in Distribution
1.4.5 Finding a Limiting Distribution by Moment Generating Functions
1.4.6 The Central Limit Theorem
1.4.7 Normal Approximations in Practice
1.5 Practice problems: distributions and transformations
1.5.1 Further exercises
1.6 Practice problems: limit theorems
2 Estimation
2.1 Introduction
2.2 Method of Moments
2.3 Method of Least Squares
2.4 Method of Maximum Likelihood
2.5 Properties of Estimators
2.6 Practice problems
3 Hypothesis Testing
3.1 Introduction
3.2 The Simple Likelihood Ratio and Most Powerful Tests
3.3 Composite Hypotheses
3.4 Uniformly Most Powerful Tests
3.5 Practice problems
4 Past examination paper
4.1 Question 1
4.2 Question 2
4.3 Question 3
4.4 Question 4
4.5 Question 5
4.6 Question 6