Properties of Laplace Transforms
2.5.4 Moment Generating Functions and Cramér Transform
Properties of Cramér Transform
2.6 Reliability and Survival Analysis
2.7 Exercises and Complements
3 Random Vectors
3.1 Relations Between Random Variables
3.1.1 Covariance
Properties of Covariance and Correlation Coefficients
3.1.2 Independence of Random Variables
3.1.3 Stochastic Order Relation
3.1.4 Entropy
Properties of Entropy
3.2 Characteristics of Random Vectors
3.2.1 Product of Probability Spaces
3.2.2 Distribution of Random Vectors
Properties of Multi-dimensional Distribution Functions
Properties of Densities of Random Vectors
Properties of Covariance Matrices
3.2.3 Independence of Random Vectors
Properties of Covariance Matrices of Two Vectors
3.3 Functions of Random Vectors
3.3.1 Order Statistics
3.3.2 Sums of Independent Variables or Vectors
Properties of Convolution
3.3.3 Determination of Distributions
3.4 Gaussian Vectors
3.5 Exercises and Complements
4 Random Sequences
4.1 Enumerable Sequences
4.1.1 Sequences of Events
Properties of Superior and Inferior Limits of Events
4.1.2 Independence of Sequences
4.2 Stochastic Convergence
4.2.1 Different Types of Convergence
4.2.2 Convergence Criteria
4.2.3 Links Between Convergences
4.2.4 Convergence of Sequences of Random Vectors
4.3 Limit Theorems
4.3.1 Asymptotics of Discrete Distributions
4.3.2 Laws of Large Numbers
4.3.3 Central Limit Theorem
4.4 Stochastic Simulation Methods
4.4.1 Generating Random Variables
4.4.2 Monte Carlo Simulation Method
4.5 Exercises and Complements
5 Introduction to Statistics
5.1 Non-parametric Statistics
5.1.1 Empirical Distribution Function
5.1.2 Confidence Intervals
5.1.3 Non-parametric Testing
5.2 Parametric Statistics.

5.2.1 Point Estimation
5.2.2 Maximum Likelihood Method
5.2.3 Precision of the Estimators
5.2.4 Parametric Confidence Intervals
5.2.5 Testing in a Parametric Model
5.3 The Linear Model
5.3.1 Linear and Quadratic Approximations
5.3.2 The Simple Linear Model
5.3.3 ANOVA
For Two Samples
One Way Model
Two Way Model
5.4 Exercises and Complements
Further Reading
Measure and Probability
Probability Theory and Statistics
Applications
Index.