*Here, you find my whole video series about Probability Theory in the correct order and I also help you with some text around the videos. If you want to test your knowledge, please use the quizzes, and consult the PDF version of the video if needed. When you have any questions, you can use the comments below and ask anything. However, without further ado let’s start:*

#### Part 1 - Introduction (including R)

**Probability Theory** is a video series I started for everyone who is interested in stochastic problems and statistics. We can use a lot of results that one can learn in measure theory series. However, here we will be able to apply the theorems to probability problems and random experiments. In order to this, we will use RStudio along the way:

###### Content of the video:

00:00 Introduction

01:20 simple example: throwing a die

02:54 Rstudio

05:17 Outro

With this you now know the topics that we will discuss in this series. Some important bullet points are **probability measures**, **random variables**, **central limit theorem** and **statistical tests**. In order to describe these things, we need a good understanding of measures first. They form the foundation of this probability theory course but we do not need to go into details. Now, in the next video let us discuss **probability measures**.

#### Part 2 - Probability Measures

The notion of a **probability measure** is needed to describe stochastic problems:

###### Content of the video:

00:00 Idea of a probability measure

01:40 Requirements

04:19 Sigma algebra

05:50 Sigma additivity

06:31 Definition probability measure

07:22 Example

08:57 Exercise about properties of probability measures

09:30 Outro

#### Part 3 - Discrete vs. Continuous Case

We distinguish **discrete** and **continuous** cases because they often occur in applications:

###### Content of the video:

00:00 Intro

00:48 Introduction of cases

02:17 Sample Space (discrete case)

02:33 Sample Space (continuous case)

03:14 Sigma algebra (discrete case)

03:36 Sigma algebra (continuous case)

03:59 Probability measure (discrete case)

05:41 Probability measure (continuous case)

07:46 Example (discrete case)

08:44 Example (continuous case)

10:38 Outro

10:57 Endcard

#### Part 4 - Binomial Distribution

Now we talk about a special discrete model: the **binomial distribution**. It occurs when we toss a coin n times and count the heads. Alternatively, we could draw n balls, with replacement, from a urn with two different kinds of balls:

###### Content of the video:

00:00 Intro

00:10 Binomial distribution

06:27 Binomial distribution in RStudio

09:40 Urn model in RStudio

14:55 Comparison of urn model and rbinom in RStudio

15:36 Endcard

#### Part 5 - Product Probability Spaces

Now we talk about **product spaces**, which will be very important for constructions of probability spaces:

#### Part 6 - Hypergeometric Distribution

The next discrete model we will discuss is the so-called **hypergeometric distribution**. It is related to the binomial distribution in an urn model. However, now we will draw **without replacement**.

###### Content of the video:

00:00 Intro

00:13 Hypergeometric distriubution

02:21 Writing a sample space for Hypergeometric function

05:42 Hypergeometric distribution for 2 colours

07:06 Hypergeometric distribution in R

10:29 Outro

#### Part 7 - Conditional Probability

In the next video, we start with a very important topic: **conditional probability**.

###### Content of the video:

00:00 Intro

00:26 Conditional Probability (definition)

05:12 example

09:09 Outro

#### Part 8 - Bayes’s Theorem and Total Probability

Now we are ready to discuss a famous theorem: **Bayes’s theorem**. We also talk about the related law of total probability and illustrate both things with the popular Monty Hall problem.

###### Content of the video:

00:00 Intro

00:17 Bayes’s theorem

01:20 Law of total Probability

04:51 Example: Monty Hall problem

09:35 Outro

#### Part 9 - Independence for Events

Next, we talk about an important concept: **independence**. We start by explaining the independence of events. First we just have two events but then we consider infinitely many.

###### Content of the video:

00:00 Intro

00:19 Visualization (Independence for events)

03:48 Definition of independence

04:52: Example (discrete case)

07:38 Continuous case

10:52 Outro

#### Part 10 - Random Variables

We are ready to introduce **random variables**. It turns out that the definition is not complicated at all. Nevertheless, we often use them to extract the important parts of a random experiment.

###### Content of the video:

00:00 Intro/ short introduction

00:56 Example (discrete)

02:57 Definition of a random variable

04:56 Continuation of the example

07:49 Notation

09:28 Outro

#### Part 11 - Distribution of a Random Variable

Next, we want to introduce the notion of **distribution of a random variable**. This is not a complicated concept but, in fact, it will be crucial in all upcoming videos.

#### Part 12 - Cumulative Distribution Function

We continue with the **cumulative distribution function** for a random variable. It is often just called CDF.

#### Part 13 - Independence for Random Variables

Now let us define the notion of **independence for random variables**. We will use the definition of independence for events for this:

#### Part 14 - Expectation and Change-of-Variables

The next concept is one of the most important ones. We talk about the **expectation** of a random variable. You also find a lot of other names for that, for example, **expected value**, **first moment**, **mean**, and **expectancy**.

#### Part 15 - Properties of the Expectation

Now let’s look at some more examples and some important **properties** of the expectation like linearity:

#### Part 16 - Variance

We continue with another important concept: **variance**. With this we can measure how much a random variable deviates from its mean.

#### Part 17 - Standard Deviation

Now we expand the notion of the variance and define the so-called **standard deviation**. We consider some examples. Most importantly, we discuss the **normal distribution** and visualize it in RStudio.

#### Part 18 - Properties of Variance and Standard Deviation

In the next video, we prove some **properties of the variance**, which can be extended to properties for the standard deviation. In particular, for independent random variables, the variance is additive.

###### Content of the video:

00:00 Intro

00:35 Properties

01:30 Variance is additive

01:50 Scaling variance

02:20 Scaling standard deviation

03:04 Proof of properties

07:48 Credits

#### Part 19 - Covariance and Correlation

In this video, we will extend the variance function from the last videos to two random variables. Therefore, it will measure how **correlated** the random variables are.

#### Part 20 - Marginal Distributions

In this video, we discuss how we can restrict random variables in several dimensions to ordinary random variables.

#### Part 21 - Conditional Expectation (given events)

In this video, we introduce a new concept for the conditional probability: the so-called **conditional expectation**. It explains what we expect as a outcome of a random variable under the condition described by an event.

#### Part 22 - Conditional Expectation (given random variables)

We extend the notion from the last video for conditions given by another random variable. This means that the, here defined, conditional expectation is a random variable.

#### Part 23 - Stochastic Processes

With this video, we start a new topic about stochastic processes, which are often used in applications. First, we give the definition and then we look at some simple examples.

#### Part 24 - Markov Chains

Most of the time, the stochastic processes we consider have some additional properties. A so-called **Markov process** does not need the past to calculate the future. The present state is enough for this.

#### Summary of the course Probability Theory

- You can download the whole PDF here and the whole dark PDF.
- You can download the whole printable PDF here.
- Test your knowledge in a full quiz.