The contents include:
Introduction, Preliminary Results, Distributions, Random Variables,
Expectation, Independence, Weak Law of Large Numbers, Borel-Cantelli Lemmas,
Strong Law of Large Numbers, Random Series, Weak Convergence, Characteristic
Functions, Central Limit Theorems, Poisson Convergence, Stein's Method,
Random Walk Preliminaries, Stopping Times, Recurrence, Path Properties, Law
of The Iterated Logarithm.
The aim of
the notes is to combine the mathematical and theoretical underpinning of
statistics and statistical data analysis with computational methodology and
practical applications. Topics covered includes: Notion of probabilities,
Probability Theory, Statistical models and inference, Mean and Variance, Sets,
Combinatorics, Limits and infinite sums, Integration.
These notes are intended to
give a solid introduction to Probability Theory with a reasonable level of
mathematical rigor. Topics covered includes: Elementary probability,
Discrete-time finite state Markov chains, Existence of Markov Chains,
Discrete-time Markov chains with countable state space, Probability triples,
Limit Theorems for stochastic sequences, Moment Generating Function, The Central
Limit Theorem, Measure Theory and Applications.
This book covers the following
topics: Basic Concepts of Probability Theory, Random Variables, Multiple Random
Variables, Vector Random Variables, Sums of Random Variables and Long-Term
Averages, Random Processes, Analysis and Processing of Random Signals, Markov
Chains, Introduction to Queueing Theory and Elements of a Queueing System.
The goal to to help the student figure out the meaning of various
concepts in Probability Theory and to illustrate them with examples. Topics
covered includes: Modelling Uncertainty, Probability Space, Conditional
Probability and Independence, Random Variable, Conditional Expectation, Gaussian
Random Variables, Limits of Random Variables, Filtering Noise and Markov Chains