Studies axioms, combinatorial analysis, independence and conditional probability, discrete and absolutely continuous distributions, expectation and distribution of functions of random variables, laws of large numbers, central limit theorems, and simple Markov chains if time permits

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Course notes

## Central Limit Theorem – MATH 4510/5510

Central Limit Theorem and little about histograms

## The Law of Large Numbers – MATH 4510/5510

The Law of the large numbers: statement, proof and how to use it

## Independent Random Variables – MATH 4510/5510

Introduction to independence for random variables

## The Joint Cumulative Distribution – MATH 4510/5510

Introduction to joint distributions

## Review for Midterm II – MATH 4510/5510

Review of all important concepts for the Midterm II

## The Continuous Uniform Random Variable – MATH 4510/5510

Definition and properties of uniformly distribution random variables over intervals

## Continuous Random Variables – MATH 4510/5510

Introduction to Continuous Random Variables

## How to prepare for Midterm II and other comments – MATH 4510/5510

Tips and comments about Midterm II

## The Cumulative Distribution Function – MATH 4510/5510

The Cumulative Distribution Function and some properties

## Binomial Distribution – MATH 4510/5510

One of the most important discrete distributions

## Variance and Standard deviation – MATH 4510/5510

Introduction to variance and standard deviation

## Expected Value of Sums of Random Variables – MATH 4510/5510

Expected value of sums of random variables and the substitution formula

## Expected value and Investments – MATH 4510/5510

Building a mathematical model to understand the impacts of deviating too much from the expected value

## Expected value of a function of a random variable – MATH 4510/5510

Formula to compute expected value of a function of a random variable

## Expected Value – MATH 4510/5510

Expected value of discrete random variables and connection with Law of Large Numbers

## Random Variables – MATH 4510/5510

General definition of random variables and examples in the discrete case

## Monty Hall Dilemma – MATH 4510/5510

Solving and discussing the famous Monty Hall dilemma

## The m points problem – MATH 4510/5510

An Important historical problem involving probability theory

## The Conditional Measure – MATH 4510/5510

Investigating the conditional probability and an example from genetics

## Conditional Probability – Part II – MATH 4510/5510

Introduction to conditional probability

## Conditional Probability – Part I – MATH 4510/5510

Introduction to conditional probability

## Midterm I – Review I – MATH 4510/5510

Comments about Midterm I and theoretical review

## Probability as a measure of belief – MATH 4510/5510

Probability as a measure of belief

## Probability measure as a continuous set function – MATH 4510/5510

Probability measure as a continuous set function. Increasing and decreasing events.

## Sample spaces with equally likely outcomes – MATH 4510/5510

Sample spaces with equally likely outcomes

## Basic Principle of Counting – MATH 4510/5510

(Generalized) Basic Principle of Counting and Permutations

## Introduction to the course – MATH 4510/5510

General comments about probability theory and our course!