Probability Theory as the Study of Random Phenomena
Probability Theory as the Study of Mathematical Models of Random Phenomena
The Sample Description Space of a Random Phenomenon
Events
The Definition of Probability as a Function of Events on a Sample Description Space
Finite Sample Description Spaces
Finite Sample Description Spaces With Equally Likely Descriptions
Notes on the Literature of Probability Theory
Samples and n-Tuples
Posing Probability Problems Mathematically
The Number of "Successes" in a Sample
Conditional Probability
Unordered and Partitioned Samples-Occupancy Problems
The Probability of Occurrence of a Given Number of Events
Independent Events and Families of Events
Independent Trials
Independent Bernoulli Trials
Dependent Trials
Markov Dependent Bernoulli Trials
Markov Chains
The Notion of a Numerical-Valued Random Phenomenon
Specifying the Probability Function of a Numerical-Valued Random Phenomenon
Distribution Functions
Probability Laws
The Uniform Probability Law
The Normal Distribution and Density Functions
Numerical n-Tuple Valued Random Phenomena
The Notion of an Average
Expectation of a Function With Respect to a Probability Law
Moment-Generating Functions
Chebyshev's Inequality
The Law of Large Numbers for Independent Repeated Bernoulli Trials
More About Expectation
The Importance of the Normal Probability Law
The Approximation of the Binomial Probability Law by the Normal and Poisson Probability Laws
The Poisson Probability Law
The Exponential and Gamma Probability Laws
Birth and Death Processes
The Notion of a Random Variable
Describing a Random Variable
An Example, Treated From the Point of View of Numerical n-Tuple Valued Random Phenomena
The Same Example Treated From the Point of View of Random Variables
Jointly Distributed Random Variables
Independent Random Variables
Random Samples, Randomly Chosen Points (Geometrical Probability), and Random Division of an Interval
The Probability Law of a Function of a Random Variable
The Probability Law of a Function of Random Variables
The Joint Probability Law of Functions of Random Variables
Conditional Probability of an Event Given a Random Variable. Conditional Distributions
Expectation, Mean, and Variance of a Random Variable
Expectations of Jointly Distributed Random Variables
Uncorrelated and Independent Random Variables
Expectations of Sums of Random Variables
The Law of Large Numbers and the Central Limit Theorem
The Measurement Signal-To-Noise Ratio of a Random Variable
Conditional Expectation. Best Linear Prediction
The Problem of Addition of Independent Random Variables
The Characteristic Function of a Random Variable
The Characteristic Function of a Random Variable Specifies Its Probability Law
Solution of the Problem of the Addition of Independent Random Variables by the Method of Characteristic Functions
Proofs of the Inversion Formulas for Characteristic Functions
Modes of Convergence of a Sequence of Random Variables
The Law of Large Numbers
Convergence in Distribution of a Sequence of Random Variables
The Central Limit Theorem
Proofs of Theorems Concerning Convergence in Distribution
Table I: Area under the Normal Density Function
Table II: Binomial probabilities