What a wonderfully clear and precise presentation! It cleared up several doubts, especially the distinction between a probability mass function and a random variable. Would be interested to see an example of a more complex problem that’s more easily solved through random variables than by probability mass function. (Incidentally - is the converse ever true, i.e. is there a problem that’s more easily solved using probability mass function rather than the random variable? Assuming that the random variable is known.)
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.
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!
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!
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!
I've been waiting for this video for a long time. I've long understood random variables at a surface level, and I've even used them to model things, but I've never had a good, solid understanding of what they are as mathematical objects. This video does what no statistics textbook or professor has done for me; namely, to give random variables a solid base for building intuition on. And the treating of them as deterministic functions on random inputs also helps me a lot. The description of the distinction between probability distribution and random variable was similarly helpful. I've long understood most of the consequences of saying "X is a Normally distributed Random Variable": it means I can calculate probabilities of X taking values between certain ranges based on the definite integration of a well-understood function. But now I think I understand what it _means_; namely, that X is a function which assigns a value to events. Perhaps events are pencils and the function the weight of the pencil. Then, if I understand the terminology, the probability distribution (which is a normal distribution) lets me measure the portion of the events where pencils take weights between some values I select. Wonderful video, and thank you for it!
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.
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!!!!)
1:35 "You should think of them as 16 different atoms." The funny thing is that is agrees with the technical definition of an atom. It's a singleton set with nonzero measure. In this case, for any x in our Omega, we have mu({x}) = 1/16.
Student: What's a X Y? Me: Well for starters it's not an X and it's not a Y. Substitute X Y for: Random Variable, Big Bang, Halting Problem, Neural Network, Elliptic Curve, Fundamental (theorem of) Algebra. Naming things is hard...
1:05 isn’t a probability space the triplet (omega, sigma, P) with sigma being the sigma field and P being the measure? I’ve always heard omega called the fundamental set.
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
yes, that's one of the most insane things in most statistical textbooks (they also confuse models with fitted models, samples and groups, etc). imo, the only way one can understand statistics is through programming, bc you can see how these abstract concepts map to operations on real data (and there are a few textbooks that do this)
Oh that's pretty cool. I am working on a video about looking at discrete distributions like vectors and doing projections on them. I was confused about RVs and PDFs in that contexts and this gives a good foundation to think about them
This is a good video to understand what s random variable is, and how the random aspect applies to the outcome of events, not necessarily a random number. However, this makes me question what random even means in math. Also, Why can functions act as variables but are not variables? I guess the underlying question is what is a variable to begin with.
@@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}.
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 )
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!
Love the video, but now you've got me thinking. If i roll a die with 4 sides, but each of these 4 sides does not have a value quite yet. Each of the sides, rolls a 4 sided die to determine their values. But wait, those 4 sides don't have a value quite yet... Let Z be the average value of this dice roll. Is Z now an actually random variable? 🤔Can we even calculate E(Z)? I think E(Z)= 4 times E( A normal 4 sided die), but im curious on your reasoning for how we can still call this Z non-random.
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.
OK, in terms of computer programming, it's not a constant, ergo it is a variable and it varies in an apparently chaotic way, ergo it is random. I call it a random variable and I call clickbait.
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
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.
Why does everyone keep saying "dice" as the singular lately? This isn't just you, it seems to be everywhere suddenly. What's going on? Did the singular "die" get replaced with the plural or something?
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
@@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
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!
I admire the effort. But I’m skeptical what purpose this simplification serves. People who would follow this simplification are most likely people who can think of random variables in the measure theoretic sense.
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.
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)
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.
Very good video, but you're use of the words x-axis and y-axis is incorrect imho. They're only an x-axis and a y-axis if the variables on them are x and y respectively. If you put events and probabilities on them then you should call them the event axis and the probability axis. Or you can refer to their visual orientation and call them the horizontal axis and the vertical axis.
This is needlessly splitting hairs over terms that ultimately makes no difference to our approach. Does Z change? Yes. What is a variable? Something that changes (randomly or otherwise). So Z is a variable. It doesn't matter whether we can also interpret it as something else. Is a dog an animal or a mammal? Exactly. As for the random part, we could have simply said that it can be random, but it can also not be random. End of video.
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
What a wonderfully clear and precise presentation! It cleared up several doubts, especially the distinction between a probability mass function and a random variable.
Would be interested to see an example of a more complex problem that’s more easily solved through random variables than by probability mass function.
(Incidentally - is the converse ever true, i.e. is there a problem that’s more easily solved using probability mass function rather than the random variable? Assuming that the random variable is known.)
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.
Student: What's a Dr. Pepper?
Me: Well for starters it is not a doctor and it's not a pepper.
In physics we have: "What is particle spin?"
"Well imagine a ball is spinning, except it's not a ball and it's not spinning"
I study maths but always avoided statistics but this video actually got me interested in learning more
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!
This is masterul. I'm starting stochastics and always get confused about what domain the variables are in etc. Best vid so far
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!
I've been waiting for this video for a long time. I've long understood random variables at a surface level, and I've even used them to model things, but I've never had a good, solid understanding of what they are as mathematical objects. This video does what no statistics textbook or professor has done for me; namely, to give random variables a solid base for building intuition on. And the treating of them as deterministic functions on random inputs also helps me a lot.
The description of the distinction between probability distribution and random variable was similarly helpful. I've long understood most of the consequences of saying "X is a Normally distributed Random Variable": it means I can calculate probabilities of X taking values between certain ranges based on the definite integration of a well-understood function. But now I think I understand what it _means_; namely, that X is a function which assigns a value to events. Perhaps events are pencils and the function the weight of the pencil. Then, if I understand the terminology, the probability distribution (which is a normal distribution) lets me measure the portion of the events where pencils take weights between some values I select.
Wonderful video, and thank you for it!
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.
Holy hell u made me realised why some prob classes touch upon measure cause a random variale is a measure
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!!!!)
1:35 "You should think of them as 16 different atoms."
The funny thing is that is agrees with the technical definition of an atom. It's a singleton set with nonzero measure. In this case, for any x in our Omega, we have mu({x}) = 1/16.
It's not a coincidence! I'm well aware of what an atom means in probability :)
My cousin in Romania is a vampire I'm pretty sure
Student: What's a X Y?
Me: Well for starters it's not an X and it's not a Y.
Substitute X Y for: Random Variable, Big Bang, Halting Problem, Neural Network, Elliptic Curve, Fundamental (theorem of) Algebra.
Naming things is hard...
Iti multumesc. Este chiar singura mea problema care m-a intepa cand invatam la probabilitati
1:05 isn’t a probability space the triplet (omega, sigma, P) with sigma being the sigma field and P being the measure? I’ve always heard omega called the fundamental set.
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
Thank you for the title of the evideo. Finally someone spoke THE TRUTH.
yes, that's one of the most insane things in most statistical textbooks (they also confuse models with fitted models, samples and groups, etc).
imo, the only way one can understand statistics is through programming, bc you can see how these abstract concepts map to operations on real data (and there are a few textbooks that do this)
The best explanation I’ve come across so far
Oh that's pretty cool. I am working on a video about looking at discrete distributions like vectors and doing projections on them. I was confused about RVs and PDFs in that contexts and this gives a good foundation to think about them
That sounds cool! Doing a projection must be equivalent to a kind of conditional expectation. Although I'm not sure of the details!
very well done explainer, thank you
Thank you!
Taking statistical estimation of dynamical systems this really helped!
“Is a function a variable? Not really.”
The lambda calculus: bet.
Wow this was a neat explanation
This is a good video to understand what s random variable is, and how the random aspect applies to the outcome of events, not necessarily a random number.
However, this makes me question what random even means in math.
Also, Why can functions act as variables but are not variables? I guess the underlying question is what is a variable to begin with.
@@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}.
Love this video!! Awesome stuff
We assign a probability to each point in the sample space Omega? How do we make sure that the sum of the sample space equals 1?
As long as they are all positive, you can always divide by the total!
Interesting angle! Thanks for sharing. **SUBSCRIBED**
Could you suggest some good books for advanced probability theory with number theory connections??
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..
Which software did you use for this animation sir?
Manim!
@@MihaiNicaMath and sir which software did u use for the slides?
great video!
Thank you!
It is a "chance function", "random variable" is a mistranslation from French.
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
I understood the interpretation of them is different but could you say a probability distribution is a special case of a random variable?
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!
Love the video, but now you've got me thinking.
If i roll a die with 4 sides, but each of these 4 sides does not have a value quite yet. Each of the sides, rolls a 4 sided die to determine their values. But wait, those 4 sides don't have a value quite yet...
Let Z be the average value of this dice roll. Is Z now an actually random variable? 🤔Can we even calculate E(Z)? I think E(Z)= 4 times E( A normal 4 sided die), but im curious on your reasoning for how we can still call this Z non-random.
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
OK, in terms of computer programming, it's not a constant, ergo it is a variable and it varies in an apparently chaotic way, ergo it is random. I call it a random variable and I call clickbait.
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
What's so hard about saying "It's a randomly generated variate from a computable distribution" or it's "a number with a computable surprisal".
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.
Good video, sir
Why does everyone keep saying "dice" as the singular lately? This isn't just you, it seems to be everywhere suddenly. What's going on? Did the singular "die" get replaced with the plural or something?
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
Use the mean-average, which if carefully maintained can approximate any random input. I'm triggered by "random is real"!
what happens with quantum mechanics? there RVs are really random I think
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!
Not to forget parallel universes, which in reality are not parallel, and not universes either! As found out by the great scientist D. Adams.
I admire the effort. But I’m skeptical what purpose this simplification serves. People who would follow this simplification are most likely people who can think of random variables in the measure theoretic sense.
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.
Why is it that (expected value of a sum) = sum (expected values).....?
@@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
It's a function.
What in math *IS* a variable? Seems to me that all names are immutably bound to some mathematical object.
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.
interesante
my hyperborean ancestors studied mathematics too
Very good video, but you're use of the words x-axis and y-axis is incorrect imho. They're only an x-axis and a y-axis if the variables on them are x and y respectively. If you put events and probabilities on them then you should call them the event axis and the probability axis. Or you can refer to their visual orientation and call them the horizontal axis and the vertical axis.
WHAT
The singular of “dice” is “die”.
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!
This is needlessly splitting hairs over terms that ultimately makes no difference to our approach. Does Z change? Yes. What is a variable? Something that changes (randomly or otherwise). So Z is a variable. It doesn't matter whether we can also interpret it as something else. Is a dog an animal or a mammal? Exactly.
As for the random part, we could have simply said that it can be random, but it can also not be random.
End of video.
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
And a continuous random variable isn't always continuous.