Preface
Chapter 1 Introduction to Probability
1.1 Introduction: Why Study Probability?
1.2 The Different Kinds of Probability
Probability as Intuition
Probability as the Ratio of Favorable to Total Outcomes (Classical Theory)
Probability as a Measure of Frequency of Occurrence
Probability Based on an Axiomatic Theory
1.3 Misuses, Miscalculations, and Paradoxes in Probability
1.4 Sets, Fields, and Events
Examples of Sample Spaces
1.5 Axiomatic Definition of Probability
1.6 Joint, Conditional, and Total Probabilities; Independence
Compound Experiments
1.7 Bayes' Theorem and Applications
1.8 Combinatorics
Occupancy Problems
Extensions and Applications
1.9 Bernoulli Trials-Binomial and Multinomial Probability Laws
Multinomial Probability Law
1.10 Asymptotic Behavior of the Binomial Law: The Poisson Law
1.11 Normal Approximation to the Binomial Law
Summary
Problems
References
Chapter 2 Random Variables
2.1 Introduction
2.2 Definition of a Random Variable
2.3 Cumulative Distribution Function
Properties of FX(x)
Computation of FX(x)
2.4 Probability Density Function (pdf)
Four Other Common Density Functions
More Advanced Density Functions
2.5 Continuous, Discrete, and Mixed Random Variables
Some Common Discrete Random Variables
2.6 Conditional and Joint Distributions and Densities
Properties of Joint CDF FXY (x, y)
2.7 Failure Rates
Summary
Problems
References
Additional Reading
Chapter 3 Functions of Random Variables
3.1 Introduction
Functions of a Random Variable (FRV): Several Views
3.2 Solving Problems of the Type Y = g(X)
General Formula of Determining the pdf of Y = g(X)
3.3 Solving Problems of the Type Z = g(X, Y )
3.4 Solving Problems of the Type V = g(X, Y ), W = h(X, Y )
Fundamental Problem
Obtaining fVW Directly from fXY
3.5 Additional Examples
Summary
Problems
References
Additional Reading
Chapter 4 Expectation and Moments
4.1 Expected Value of a Random Variable
On the Validity of Equation 4.1-
4.2 Conditional Expectations
Conditional Expectation as a Random Variable
4.3 Moments of Random Variables
Joint Moments
Properties of Uncorrelated Random Variables
Jointly Gaussian Random Variables
4.4 Chebyshev and Schwarz Inequalities
Markov Inequality
The Schwarz Inequality
4.5 Moment-Generating Functions
4.6 Chernoff Bound
4.7 Characteristic Functions
Joint Characteristic Functions
The Central Limit Theorem
4.8 Additional Examples
Summary
Problems
References
Additional Reading
Chapter 5 Random Vectors
5.1 Joint Distribution and Densities
5.2 Multiple Transformation of Random Variables
5.3 Ordered Random Variables
5.4 Expectation Vectors and Covariance Matrices
5.5 Properties of Covariance Matrices
Whitening Transformation
5.6 The Multidimensional Gaussian (Normal) Law
5.7 Characteristic Functions of Random Vectors
Properties of CF of Random Vectors
The Characteristic Function of the Gaussian (Normal) Law
Summary
Problems
References
Additional Reading
Chapter 6 Statistics: Part 1 Parameter Estimation
6.1 Introduction
Independent, Identically, Observations
Estimation of Probabilities
6.2 Estimators
6.3 Estimation of the Mean
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