Probability Theory 5 | Product Probability Spaces

[@jaimelima2420]

Robert Gallager, the famous MIT professor, usually warns engineering students to not go too deep in Measure Theory while studying statistics. He says that he 'lost' a few good people trying to go into this adventure. They go and never come back and we never hear anything about them back in the engineering field...

[@kim8u96]

This is probably a trick by some mathematicians to get people to fall in the pure mathematics hole. Like: "aw, you use the differential as if it were a number? Let me take revenge now."

[@jaimelima2420]

@@kim8u96 "Once upon a time there existed a mathematician who was very 'mean', but he/she was changed into a better person after having to handle with a pathological distribution."

[@pranavdeshpande4942]

Lol that's good advice, I was trying to do that and it didn't end well....

[@euniceng_glorytohk]

Just signed up the membership! I don't see any math content creator or online math courses covering measure theory or probability theory. I am a new graduate statistics student with very weak math background - before this program, I was a business student. I signed up a class called "probability" and it turned out to be awfully abstract - theories everywhere! I was so desperate that I searched multiple online education platforms for courses with similar coverage to give myself more context. And your youtube videos are the only thing relevant!!! And it is free! I am about to finish your measure theory series and now I am starting your probability series! I am going to share this with my friends with same struggle!!! Also I am not having an easy time in my real analysis this semester. I am going to your another series of analysis later. (I have no idea why I selected that many courses with such large amount of theories and seem-to-be not job-seeking-useful. )

[@brightsideofmaths]

Thank you very much! It's very appreciated that you share my videos with your friends. More viewers motivate me even more to produce more math content :)

[@hsujerry7231]

Very good video, the last example also can be used to explain the probability measure on an event in a filtration!

[@carlosv.ramirezibanez1104]

Great video, this really helped me for the concept of “random vector”, thanks!!

[@brightsideofmaths]

You're very welcome!

[@sharonarandia3630]

Your videos are awesome, thank you.

[@daliagutierrezvalencia1196]

I really, really appreciate your videos! Thank you very much! 💓

[@brightsideofmaths]

Glad you like them!

[@riccardobova6738]

Hello! Just a question: what is the relation between this definition and the concept of indipendence? When we say that the probability of cartesian product of two sets is the product of probability... are we implicitly assuming some sort of indipendence?

[@brightsideofmaths]

In some sense, this is correct, but we will talk about this later in more detail.

[@saimasiddiqua7532]

Wow you describe it so easy way. It helps me to imagine .Thanks a lot. Such as TALAGRAND Inequality.

[@brightsideofmaths]

Thanks and welcome

[@markwilliams1555]

Great video. Keep up the good work! It's very much appreciated. I haven't had much a chance to look through the playlist but do you have anything on the monotone class lemma?

[@JojiThomas7431]

Incredible

[@ivanzhao8705]

Hey I love your lecture is there any channel to donate ?

[@brightsideofmaths]

Thank you very much. Paypal and Steady links you find in the description. Any support is very appreciated :)

[@Hold_it]

Beautifull

[@ivanbooroovooy6775]

I don't understand, why everyone is happy about these videos. I was trying to indulge again into statistics and probability using this material and was very confused. First, what caused the confusion for me is how you define the possible outcome for an experiment with outcome being a continuous random variable (throwing a point on the interval). This was confusing in the previous video too, but since it was the only mistake - I just closed my eyes on this and went further. But here I see how you further try to calculate the probability for this experiment, and this is completely unforgivable and does not make any sense. The probability of the outcome of any experiment of throwing the point on any interval is always infinitely small due to the continuous nature of the interval. The right way of defining the experiment is to say what is the probability of throwing the point BETWEEN a and b on the interval - and further calculations are well-known. Second thing is how you define the probability for both discrete and continuous spaces. I do not understand, why do you take both sum and integral on the subset of power set or on Borel algebra instead of sample space omega. Literally in the previous video you have explained, what may a subset of power set consist of - and there may be very many elements into such subset so that the propabitily function as you defined it would in only few cases equal to 1 when summed by the elements of the power set (however on such sum should always be by the definition). On the contrary, would you take the sum on sample space - the smallest possible outcome will be 0, and the biggest - 1, just as it has to be. Hence, a certain mistake there. Third is the integral for the probability for the point on the interval experiment. What on earth is this? Somehow distribution function turns out to be 1/2 (which might not always be the case, here it is the constant distribution, but you don't say it, and it MAY be different) and, again, the integral is taken on Borel algebra?? If about the first mistake I am very sure - about the second and third thing I just cannot be as sure because I don't understand what you are trying to calculate at this point. Anyway, for all people who is going to see this video - this is how would you do it: For the throwing a point on the interval [-1, 1], assuming you have a distribution function of 1/2, you would take an integral on some part of sample space for which you want to calculate probability, for example on [0, 1] or [0, 0.5], etc. which is 1/2*length of your interval. What did the person running this channel mean - I don't understand, hopefully at least he understands that himself!

[@brightsideofmaths]

I find this comment very rude. I try to teach the concept of a probability measure, which also makes sense in the continuous case. I am happy to help for any problems in understanding but attacking me in the very first sentence does not help.

[@ivanbooroovooy6775]

@@brightsideofmaths Okay, I am very sorry for the rude part, I will edit this right after writing this answer. However, I believe, I have in a very discrete manner explained all parts which seemed controversial to me, especially with the fact that probability theory is not controversial at all if you use the most common Kolmogorian axioms, on which statistics are built entirely. And you did not either agree or disagree with me yet, which also lightens some doubts.

[@brightsideofmaths]

I can answer some questions: (1) The probability of hitting 1/4 as a point in the interval is 0. There you are right :) (2) The definition of the probability for this experiment was discussed in part 3. And yes, there we also wrote down the probability of throwing the point BETWEEN a and b on the interval. (3) Integrals and probability measures are always defined for sigma algebras. That is why I mention the power set and the Borel sigma algebra, see part 2. (4) The distribution function for the interval experiment is 1/2 because the length of the interval is 2 as you also mentioned. (5) The Kolmogorov axioms are satisfied with the definition of probability measures as we have defined them in part 2. @@ivanbooroovooy6775

[@ivanbooroovooy6775]

@@brightsideofmaths (1) Well, about the first one, as I said earlier, I was sure, and I still do not understand why have you decided to calculate probability of the experiment outcome through defining the measure if by your own definition it is straight equal to 0. It is a very bad example, maybe even the worst as you cannot even represent it on the picture. (2) Okay, I did see how you defined the probability of throwing the point between a and b, just as I said in my previous comment "with eyes closed" and that's why. I got through this part without any confusions because I already know, why the probability of getting into a point is zero, but what you said is that the probability of getting into a point is 0, and THEN we have to take an integral because of that, so you postulate this. There is no such postulate anywhere on earth, and the only reason why the probability of getting into the point is 0 is the definition of the integral with the same limits, not because "it doesn't make any sense". (3) So you write that integrals and probability measures are always defined on sigma-algebras - that is great and true. But why? Why would you not simply define it on the sample space (for the continuous case) and use cumulative distribution instead of using measure? Because - if we are talking about always - you are the first person in my life among many, including Kolmogorov's students who taught me (I am not saying they are better at teaching or that I am a nice and a smart student, basically, that is why I am here), who prefer to define probability distribution using measure. Does it have anymore benefits than just using a much easier and more straightforward CDF? (3*) I also asked about the integral on sigma-algebra, quoting "what on earth is that integral?". I managed to find Lebesgue-Stieltjes integral which is used to integrate sigma-algebras, since nor basic Riemann, neither simple Lebesgue cannot integrate this. Eventually, I found a video about all these uneasy stuff on your measure theory playlist. I assume, you expected people to instead of watching this playlist go first through all you measure theory playlist to get at least a basic understanding of what is the integral you write (when you could possibly just point it out in four word inside the video, namely "notice Lebesgue-Stieltjes integral"). However, after all the question still persist, why would you define the probability for the continuous case this way when both at the lectures and in books (talking about walpole/myers, for example) CDF approach is preferred? Are there any particular benefits of doing that?

[@brightsideofmaths]

(1) The measure is not equal to zero. Therefore, it's a very good example: even while the outcome of each singleton is zero, the whole measure is not zero. (2) I have a whole series about Measure Theory for this kind of stuff. In this series, we concentrate on the discrete and the continuous case. So I don't have to go into the details here. (3) The approach with measures is more general and covers also the mixed cases nicely. (3*) It's always a gamble which knowledge to require when teaching mathematics. If you don't like this approach here, it's not a problem. Maybe it's not helpful for you to understand probabilities. However, it definitely helped me. You don't have to use my videos if they just confuse you.