Introduction to Probability and Random Variables

Preface
Contents
Chapter 1: Experiments, Sample Spaces, Events, and Probability Laws
	1.1 Fundamental Definitions: Experiment, Sample Space, Event
	1.2 Operations on Events
	1.3 Probability and Probabilistic Law
	1.4 Discrete Probability Law
	1.5 Joint Experiment
	1.6 Properties of the Probability Function
	1.7 Conditional Probability
Chapter 2: Total Probability Theorem, Independence, Combinatorial
	2.1 Total Probability Theorem, and Bayes´ Rule
		2.1.1 Total Probability Theorem
		2.1.2 Bayes´ Rule
	2.2 Multiplication Rule
	2.3 Independence
		2.3.1 Independence of Several Events
	2.4 Conditional Independence
	2.5 Independent Trials and Binomial Probabilities
	2.6 The Counting Principle
	2.7 Permutation
	2.8 Combinations
	2.9 Partitions
	2.10 Case Study: Modeling of Binary Communication Channel
	Problems
Chapter 3: Discrete Random Variables
	3.1 Discrete Random Variables
	3.2 Defining Events Using Random Variables
	3.3 Probability Mass Function for Discrete Random Variables
	3.4 Cumulative Distribution Function
	3.5 Expected Value (Mean Value), Variance, and Standard Deviation
		3.5.1 Expected Value
		3.5.2 Variance
		3.5.3 Standard Deviation
	3.6 Expected Value and Variance of Functions of a Random Variable
	3.7 Some Well-Known Discrete Random Variables in Mathematic Literature
		3.7.1 Binomial Random Variable
		3.7.2 Geometric Random Variable
		3.7.3 Poisson Random Variable
		3.7.4 Bernoulli Random Variable
		3.7.5 Discrete Uniform Random Variable
	Problems
Chapter 4: Functions of Random Variables
	4.1 Probability Mass Function for Functions of a Discrete Random Variable
	4.2 Joint Probability Mass Function
	4.3 Conditional Probability Mass Function
	4.4 Joint Probability Mass Function of Three or More Random Variables
	4.5 Functions of Two Random Variables
	4.6 Conditional Probability Mass Function
	4.7 Conditional Mean Value
	4.8 Independence of Random Variables
		4.8.1 Independence of a Random Variable from an Event
		4.8.2 Independence of Several Random Variables
	Problems
Chapter 5: Continuous Random Variables
	5.1 Continuous Probability Density Function
	5.2 Continuous Uniform Random Variable
	5.3 Expectation and Variance for Continuous Random Variables
	5.4 Expectation and Variance for Functions of Random Variables
	5.5 Gaussian or Normal Random Variable
		5.5.1 Standard Random Variable
	5.6 Exponential Random Variable
	5.7 Cumulative Distribution Function
		5.7.1 Properties of Cumulative Distribution Function
	5.8 Impulse Function
	5.9 The Unit Step Function
	5.10 Conditional Probability Density Function
	5.11 Conditional Expectation
	5.12 Conditional Variance
	Problems
Chapter 6: More Than One Random Variables
	6.1 More Than One Continuous Random Variable for the Same Continuous Experiment
	6.2 Conditional Probability Density Function
	6.3 Conditional Expectation
		6.3.1 Bayes´ Rule for Continuous Distribution
	6.4 Conditional Expectation
	6.5 Conditional Variance
	6.6 Independence of Continuous Random Variables
	6.7 Joint Cumulative Distribution Function
		6.7.1 Three or More Random Variables
		6.7.2 Background Information: Reminder for Double Integration
		6.7.3 Covariance and Correlation
		6.7.4 Correlation Coefficient
	6.8 Distribution for Functions of Random Variables
	6.9 Probability Density Function for Function of Two Random Variables
	6.10 Alternative Formula for the Probability Density Function of a Random Variable
	6.11 Probability Density Function Calculation for the Functions of Two Random Variables Using Cumulative Distribution Function
	6.12 Two Functions of Two Random Variables
Bibliography
Index