# Probability Theory

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:

#### 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
09:25 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
01:50 Scaling variance
02:20 Scaling standard deviation
03:04 Proof of properties
07:48 Credits

#### 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.