Probability Theory 3 | Discrete vs. Continuous Case

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This is my video series about Probability Theory. I hope that it will help everyone who wants to learn about it.
This video is about probability theory, also known as stochastics, stochastic processes or statistics. I keep the title in this general notion because I want cover a lot of topics with the upcoming videos.

Here we talk about discrete and continuous models. In the general theory both cases are included. However, it makes sense to distinguish between them because in applications one often has the one or the other case. Therefore, it is helpful to know the differences.

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

#ProbabilityTheory
#Analysis
#Calculus
#Mathematics

I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[brightsideofmaths]

This a re-upload of this part because the original video had a small mistake that Timothy Zhou found. Thanks!

[deltax]

i love how easily digestible these videos are. These are really going to help me in my Distribution theory and inference class at FSU

[rob]

As the series progresses will you cover (for completeness) the Essential Supremum?

[theoharkes]

Hey Julian, first off all thanks for your videos (especially on measure theory); they help me a great deal with my master thesis.
I was wondering if you are planning to make a video on the mixed / other cases any time soon? I'm constantly running in to problems when I have continuous distributions that overlap with Dirac delta's, and it bugs me that I don't know if I'm handling them in a decent way.

[bebert]

Thanks a lot for your videos. I watched the probability and measure theory playlists and it helped me a lot. I was wondering if by any chance you planned on developing the mixed case?

[DynaML-basic]

Hey Julian, which software you use for your presentations? It think like a board and I need it to do math and data science lessons. Can you help me?

[ecg]

I am sorry, I got lost when you set f(x) = 1/2 in the example of the continuous function... Could you elaborate a little more on that? Thanks!!

[sinx]

I assume that we could say the probability measure in the discrete case is also an integral over the sample space, but with respect to the counting measure instead of the Lebesgue measure.
Also unless I'm missing something, the definition of the pdf as a measurable map from the sample space to R seems to be the same definition as a (concrete) random variable, but just with the additional requirement of having an integral of 1. Does this mean that pdfs are random variables, and can we treat them as such? Is there such thing as a distribution or expectation of a pdf?

[evionlast]

I'm assuming that before conditional probability the subject of dependence and non equiprobable events will be addressed a little further?

[jonasw]

In the descrete case, when Omega contains countable infinity elements: Isn't the probability of a single event also 0? In your example if Omega = naturalNumbers isn't the probability to "draw" a 5 equal to 0?

[premkumar-soff]

I am going to follow your 18 videos but want to know what is the general pattern of probability at undergrad and gradute level probability. Mainly i came to know that probability is studied as an area of mathematical analysis from undergraduate level. Previously here in india we studied probability basic definitions and bayes theorem at high school level algebra. But i dont know how to follow the pattern that takes to measure theoretical level as i already have done graduate real analysis course called measure theory and lebesgue integration etc. Any specific pattern of chapters and books you suggest.

[BatuhanBayr]

Is it true that "probability of a singe point is just zero because we have uncountably many points on dart"? For this reason we must deal with probability for regions.

[alfabetet]

Thanks for doing this, i have Einführung in die
Wahrscheinlichkeitstheorie und Statistik right now. But my German isn't too good. So this really helps.

[lui]

The Videos are so great, thank you! Maybe they could upload in german too? ... That would be awesome!!