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@@tempusername-l5d Typically, a variable is a fixed (but unknown) quantity, like x in the equation 2x+5=7. You can apply operations to it to e.g. solve for x to get x=1. The point is that a random variable is not like that but all the operations still work, so you can use them like you use variable. E.g. if you want the event that {2X+5=7} this is the same as the event {X=1}.

Manim!

@@MihaiNicaMath and sir which software did u use for the slides?

It totally insane to me that it's possible to take a measure theory class but not mention the connection to probability. (My first measure theory class was like this! I thought I hated measure theory!!!!)

That sounds cool! Doing a projection must be equivalent to a kind of conditional expectation. Although I'm not sure of the details!

Unfortunately, no! It just means waiting for a sequence of two counflips is a lot trickier than waiting for one

Do you just mean because the probability space has finitely many point (as opposed to being an arbitrary set/measure)? My feeling is its easier and helpful for undergraduate students to first understand the sinpler case before moving onto the general case

@@MihaiNicaMath Maybe it's just me. But I think the idea of a random variable as a function only feels natural once you can appreciate the requirement that the function be measurable. And to appreciate the requirement, we need to appreciate what a probability triple is. In my opinion, once you skip over the "technical details" of sigma algebras and probability measures, the idea of random variable as a function invites complication without the reward of greater sophistication and generalization. With this video, those who would understand the measure theoretic definition may feel that something is missing. And those who would never understand the measure theoretic definition probably just got more confused. But I must admit I have grown rather cynical about most people's capacity for abstract thinking.

As long as they are all positive, you can always divide by the total!

Much like x=x+1 means something different in programming and in math, so too do the words "variable". This video is about the math definition of random variable! (And if course the interpretation as "these numbers are random" is used a lot in computer programming!)

@@MihaiNicaMath I know, but I am not a mathematician - and you managed to nerd-snipe me into watching for a bit - until I realised this was not going to be about coding. Doh! Anyhoo, you got plenty of great feedback from people who came here for the math. 🙂

@@smudgerdave1141 Thanks for the feedback! If you like randomness in computer programming, you might like my earlier video about using a GPU to estimate Pi by simulating a bunch of noodles ruclips.net/video/po_pmPrO2YY/видео.html

It's the same thing actually! In classical mechanics, randomness comes from your lack of knowledge of the exact state, so the probability space can be thought of as being over the set of classical states. In quantum mechanics, the state alone is not enough to determine the outcome, so the probability space is larger than just the state space. But the fact that the random variable is a function on the probability space is the same in both!

Guess it’s easier to say and also it’s close to death.

In the UK people (including old people) generally say dice. So I don't think it's a recent change. Although people do say "the die is cast", but most people don't actually know what it is referring to xD

It's even worse than that, back in my day they would say "alea iacta est"! Kids these days....

@@tommyphillips1030 Oh interesting. I always thought "the die is cast" referred to the process of die casting, like pouring molten metal into a form until it sets. It seems you're right, though, it means throwing a die. That doesn't seem very final to me, though. Just pick it up and throw it again?

@@severoonI believe it’s referring to games of old where you would wager for example your cow or pig or firstborn child on the outcome of a cast of a die, please correct me if I’m wrong

Unbound variable: an arbitrary symbol/name in a mathematical expression. This expression can be evaluated with the variable bound to a specific mathematical object (eg. a Real Number). For any unbound variable, it is usually implicitly understood that is can only be bound to mathematical objects of some specific type T, and only operations thats make sense on objects of type T can be performed on the unbound variable. Around 11:15 when he says you can "use it is a variable", I think he means you can use it as a variable of type Real Number.

Yes exactly! So like even though Z is a function (and it's incorrect to say it has a particular value), you can do any operation that you can do on real numbers to it.

Some of my earlier Summer of Math Exposition videos show problems with tricky ways to manipulate the random variables to get the answer: The ABRACADABRA theorem ruclips.net/video/t8xqMxlZz9Y/видео.html and there is another about estimating Pi with spaghetti ruclips.net/video/e-RUyCs9B08/видео.html . For many "standard" problems, using the probability distribution is the most straightforward way to go, which is why this is the main way people are taught about random variables.

Yes: in general what you said is correct. For discrete spaces with finitely many elements, you can just make sigma all the subsets so it's "trivial" in some sense, and Omega, P is all you need

How is it in the original french?

​@@victorscarpesit's random variable, not sure what he meant

@@NC-hu6xdmaybe a difference between the direct translation and the actual meaning?

@@nobody08088 The french is "variable aléatoire" which directly translates to random variable. I dont understand his comment nor the 20 upvotes

Well the actual point of the video is to give a mathematical definition of a random variable so we can do math and understand the last 100 years of literature on the subject :) I'm sorry it was so upsetting for you

"Does Z change? Yes." No! You can have many different samples of Z, but Z itself is fixed. That was the point of the video. The machinery is general enough that you can do probability without necessarily involving a 'time' aspect.

@Lolwut You have low understanding and appreciation of the foundations of probability theory based on your terse “explanation”. I would not go with you if I needed help proving a theorem

And you clearly didn’t watch the end of the video where he shows how one approach easily got the expectation of the binomial distribution while the other approach was more tedious. There are different approaches all based on solid mathematical theory. Rigor is vital

A surprising amount of reading math is constantly asking yourself: "wait what type of thing is this?". If you can keep all the domains/ranges/functions straight you are halfway there!

It's not a coincidence! I'm well aware of what an atom means in probability :)

Indeed, it's easy to say that, and that's pretty close to the "it's a random number" idea that is a good 1st level understanding of what an RV is. It's just that this is NOT the mathematical definition of an RV. The video has the details!

​@@MihaiNicaMathSince I have had a long career designing RNGs and trying to bridge the gap between formalism and physical design for nondeterministic behaviour in the RNGs I think of the randomness coming part from ignorance of state (HILL entropy) and from quantum uncertainty in the generation of bits) so in that perspective the randomness appears at a well defined point. The distribution comes from analysis of the source and post processing.

I think it's misleading to say it's a special case. Two things I would say: 1. Every RV *has* a probability distribution. (Described in video) 2. Given a probability distribution, you can create an RV that has that distribution. (Set \Omega to be the set of values it takes with P(\omega) given, and then make Z(\omega)=\omega). This is sometimes called the "canonical" match between an RV and a distribution. I didn't mention this in the video!

You are right: should be less or equal there. (Or else do min(0.9 EPs) to make it work

@@MihaiNicaMaththanks for replying! im struggling with my calculus homework, good thing there’s videos like this that can help me, keep up the good work!

​@@Arthooooo-Yuan You can do it! Don't give up :)

That's wonderful to hear!

@@victorzurkowski2388 it was proven in the video (for the finite and discrete case)

A couple things: 1. "Disjoint" and "independent" sound similar but actually mean completely different things. Don't mix them up! 2. The proof works whenever Z=X_1+X_2...you don't need any other conditions or independence of anything else. All you have to do is rearrange the sum in the definition of E(X). (For non-discrete distributions it's still true but proof is slightly more complicated!)

@@MihaiNicaMath it's also not true in the infinite (but still discrete) case, unless you come up with a canonical order for the summands (if it diverges, you have Riemann rearrangement theorem to deal with, and yet these scenarios may appear when doing game theory thought experiements)

@@MagicGonads yes, in the infinite case you need an extra condition on the finiteness of the expected value!

@@MihaiNicaMath you are very correct

yet it's not less clear since in the case of saying "die" it sounds the same as "die" or "dye"

I agree: I decided years ago it's always clearer to say "dice"!

@@MihaiNicaMath Well, you know the old expression: “Never say die!”

​@@malvoliosf Haha yes exactly!

I like the books "a first look at rigour probability" by Rosenthal and "probability with martingales" by Williams (featured in my other video: ruclips.net/video/t8xqMxlZz9Y/видео.html )

@@MihaiNicaMath Thanks a lot..

Thanks so much for the kind words, I'm glad you liked the video! Another fun example I've heard: go to the library and look at the first book someone walks out with. The probability space is the set of all possible books, and you can make many random variables: how many pages is this book? How much does it weigh? What color is it?

If you're interested in the more robust formulation of random variables you could read about probability using measure theory, you'll get a few interesting insights which this video describes.

Each side has a value of E(4 sided die) E(Z) = 4 * 1/4 * E(4 sided die) 4 sides * probablity of a side * value of the side To understand this simply, to determine the result of rolling the die, roll a 4 sided die to determine what is on the face. But that happens no matter what you roll on the original die

​@@BryanLu0 You're right. Each die face would tend to the E(Z) so then the total value would tend towards E(Z). Not 4 times E(Z). Thanks for the correction! Unrelated to your comment, because I was thinking about the problem again: Here is a different example to illustrate the point. At first have the normal 4 sided die with the middle two numbes becoming negative. . Each recursive new die does this to the faces currently on the die before it: Multiply by a random real number, from 1->infinity. Random number=5 So 1,2,3,4 become 5,-10,-15,20 Repeat again. lets say 10 50,-100,-150,200 etc. Now when it does get back up to the original dice, the one we actually care about, what will the E(Z) be? It will be 0. Yet is Z a RV. that isn't a RV. or is it? Are these "R.Vs" _random_ or not? Which im not sure is or isn't "random". I think it still isn't really random. But it also kind of feels like any such dice are actually random. Because Z can be end up being anything, even without infinite recursion. Though, then the definition of "choose a random number" gets called into question. How does one choose from 1->infinity? Idk, but as a thought expirement, i'm curious what the counter-claim would be such that this "Z" is actually essentially determinant.

@@elunedssong8909 E(Z) = 0 doesn't mean Z is determinant/not RV. E.g. E(normal distribution) = 0

I had a similar experience! I was scared away by early courses, and didn't get into it until my PhD. The problem is really that statistics is too useful so there are lots of sources that try to make it accessible and end up hiding the details!

@@MihaiNicaMath engineers ruin everything

Prob theory and stats overall are very cool areas! It can be shady sometimes because it'll be "simplified" and it could make you feel like things are just happening sometimes. After intro to prob, this happens less and less, making these areas more and more fun! It's a lot of function theory which great!

I was first introduced to statistics in high school and it was incredibly boring. In uni I really enjoyed probability theory, it’s a different beast. Maybe it’s just me but getting to see the underlying mathematical machinery is really cool. You learn things like the difference between impossible and probability 0, the fact that there are many different types of convergence with different implications, and of course the Central Limit Theorem.

@@adam_jri dude that's so awesome that you enjoyed prob theory in uni! It only ever gets better! If you had fun with prob and enjoy applications as well, stats can be quite a lot of fun! And handy!