What is the average distance of two points in a disc? (PART 1)
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- Опубліковано 13 чер 2024
- This seemingly simple question actually encompasses very rich connections between different topics in mathematics, including statistics / probability theory, Jacobian / multivariable calculus, and even differential equations.
This video is the first part in a video series tackling this question, which tries to make the problem more mathematically precise by breaking down the words "random" and "average", which can have precise mathematical definition, using probability density function, and the concept of mean of a function.
The "joke" in the beginning of the video is inspired by (read: "copied from") CGPGrey's "How many countries are there?" • How Many Countries Are... ,
Tom Scott's "How many languages are there?" • How Many Languages Are... ,
and Tom Scott's "The Never-Used Road Where The BBC Crash Cars" • The never-used road wh...
Even though this problem "highlights the unity and utility of the undergraduate mathematics curriculum" (from the paper below), I would assume you know nothing, so don't worry if you are not in university / have a degree in mathematics! If you are in college / university, hopefully the first few videos can be a nice revision and application of the concepts, and possibly a new perspective on the concepts.
The paper that I am following (very readable, for an undergraduate at least):
www.tandfonline.com/doi/abs/1...
I do notice that MindYourDecisions made a similar video ( • VERY HARD Puzzle: What... ) a few years ago but for a unit square instead. I still make this video series because (1) the unit disc version is much harder to tackle in the sense that we are not even attempting to evaluate the integral, and (2) Presh's video seemed to pull pdf's and Jacobian out of nowhere, which might be confusing to people who have not gone to college to study mathematics, and genuinely quite a different level of difficulty from his other videos, so I am going to actually explain what those are.
Thanks to all my subscribers, because this channel has grown a lot since the beginning of the year, and this cannot happen without your support! Merry Christmas / happy holidays! We shall see next year, which hopefully will be better :)
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
Social media:
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For my contact email, check my About page on a PC.
See you next year!
Merry Christmas / happy holidays! Thank you to all of you supporting this channel! Hope that the next year will be better. A few "important" remarks from me because most people prefer to read comments rather than the description:
(1) The "joke" in the beginning of the video is inspired by (read: "copied from") CGPGrey's "How many countries are there?" ua-cam.com/video/4AivEQmfPpk/v-deo.html,
Tom Scott's "How many languages are there?" ua-cam.com/video/qYlmFfsyLMo/v-deo.html,
and Tom Scott's "The Never-Used Road Where The BBC Crash Cars" ua-cam.com/video/nQOroZNr_fw/v-deo.html.
(2) Around 6:28, it might be a little confusing - the gist is that we are implicitly assuming that when we say "average", we sample points across the domain using a constant pdf. If we sample the points from a different pdf, then the "average" would be some sort of "weighted average". It does not really matter too much to what we will be considering later on in the series though. I apologise for this. The excuse is that I really want to (a) explain why the two concepts are related when seemingly only the average is needed to explain the quadruple integral, and (b) get this video out before Christmas.
(3) I do notice that MindYourDecisions made a similar video (ua-cam.com/video/i4VqXRRXi68/v-deo.html) a few years ago but for a unit *square* instead. I still make this video series because (a) the unit disc version is much harder to tackle in the sense that we are not even attempting to evaluate the integral, and (b) Presh's video seemed to pull pdf's and Jacobian out of nowhere, which might be confusing to people who have not gone to college to study mathematics, so I am going to actually explain what those are. [I am not intending to start any drama here, please.]
(4) Even though this problem "highlights the unity and utility of the undergraduate mathematics curriculum" (from the paper in the description), I would assume you know nothing, so don't worry if you are not in university / have a degree in mathematics! If you are in college / university, hopefully the first few videos can be a nice revision and application of the concepts, and possibly a new perspective on the concepts.
(5) As always, do consider logging your math levels here: forms.gle/QJ29hocF9uQAyZyH6, so that I know your math levels to adjust the pace of future videos. For example, I would assume that you know basic differential and integral calculus (high-school level) because most people replying say they do. The more people doing this, the more representative the sample is.
Merry Christmas 🎄☃️
Me throughout this video: "Oh I know PDFs! oh... nevermind. Oh! I know Cartesian and Polar! oh... nevermind." I love this channel
Haha I am sure you do know pdfs and Cartesian and polar coordinate systems very well, just that in some other situations where we do need to use them, it can seem out of place!
Ye, he stopped short of saying how integrating the pdf over an area gives the probability of an observation falling in said area. (You could say pdf is a derivative of probability wrt an area/region).
So just saying the pdf is probability divided by area might feel like being given a puzzle piece without acknowledging the rest of the puzzle.
I dunno, overall it was easy to follow..
Came here from Dream, but I’ll be staying for the math.
Thanks for staying :) The next video is going to be about probability and statistics, so I am sure you will like it anyway :)
Same
I have little knowledge about probability but this video explained everything perfectly, greetings from Spain :D
Thank you so much!
I'll be a bachelor in mathematics in about a year (hopefully). I just discovered your channel and your way of explaining these concepts are amazing. Your channel will grow rapidly for sure. All the best, and hope for a better 2021 :)
Thank you for the kind words! This really warms my heart :)
@@mathemaniac Really as expected , people will realize your potential for creating a fun side of mathematics sooner or later 😉
Keep it up sir 😁
Maybe you would remember me 😁
I am one of your earliest fans
#richreformer😁
This video is really well made and very clearly explains some concepts that have perplexed me for a long time! I want to say thank you!
Thank you!
thank you a lot for all your work! I am really looking forward to watching the complete video series! (3b1b started a 3 video series about probability but never released the third ): ) Also, congratulations for your 2800 subscribers and hope many more people discover this fantastic channel next year! :)
Great video, and Merry Christmas!
Thanks! Hope you have a great Christmas as well!
Literally the best video ever! Answer given in 30 seconds!!
Please make more on probability!!!!!!! This video is awesome!!
Then you will probably love the entire series. Sadly, the next few videos will be out next year because I do want to rest for the rest of the year :)
Just found out about you. Quality presentation and content. Deserves to grow. Cheers
Thanks so much for the compliment!
This video is SO underrated!
very well explained.
Thanks for the funny start, I got the flu so I needed that
Hope you recover soon!
Nice video to a very interesting problem, looking forward to the remaining parts. Do you plan to make a video on how you make these animated videos at some point?
Thanks so much for the appreciation! I will probably reveal how I make my videos in the future - see the description.
very good presentation! your channel would get big soon hehe
I love your content so far. Just subscribed. My only recommendation is to title videos something that entices the audience to want to know more about the topic. (This video does a fine job at that, but overall, I don’t think I would not have found this channel without the video you did on Dream and I happen to be a physics student who also likes Dream so...)
Just to add, I understand if views are not a priority, but with the quality and effort you clearly put into your videos, I think many more people would benefit from it. I’m not trying to be mean when I say this. I just like the content I’ve seen so far to think more people would like to see it too
Yes, I do know that the video topics aren't the most interesting to a lot of people, but I do also want to make videos that I myself find interesting, which obviously doesn't overlap that much with that of the general public... so there would have to be some sort of compromise there...
I did not mean to criticize the content you chose to cover so far. It’s just the titles for a lot of videos are cropped because they are too long. Also, probably less important for people like me who like math for math’s sake, but very important for those who are still developing their mathematical sense of beauty, I think the viewer can often not tell what the video is about until they click on it. I and probably others trust your sense of mathematical beauty enough to click on your next videos, but you might want to try to intentionally title them sometimes with impressive achievements you can obtain using the math, or explicitly hard or beautiful problems you can solve. I understand the last part is probably hard and not doable with every topic and video, but keeping the title short is probably achievable by relying more on the thumbnail. I might be wrong, but I’m sure you can try it out. there are some apps that will switch thumbnails and titles multiple times for you and see which perform better (You may already know about them)
@@mathemaniac I understand I might be wrong, and I apologize if I overstepped. Overall, I see myself watching your videos regularly as long as you choose to keep publishing, and I hope you get the same enjoyment out of making the videos as the audience gets out of watching them
Oh no I am genuinely taking your words as compliment! It's just me giving a little rant haha.
Obviously want to try to title it so that more people want to click on it and enjoy the mathematics, but I also think it might not be possible. I'll see what I can do :)
Good Job
Crap! Merely with the tittle you gained my subscription... thanks to you man, great content!
Thanks!
Great videos and very well produced! Just got this channel recommended from the dream vid XD. BTW do you think I should consider math major I don’t really do good at math olympiads and I’ll have to start at a greek uni since I live here or something safer like engi ?
Thanks for the appreciation! I don't really want to offer career advice to a total stranger on the internet, because I don't know your situation, and it is your big decision. I highly doubt being bad at math Olympiad will impair your chance of success in the admission process, but I'm not entirely sure, so do take some time to do research for your own "dream uni".
@@mathemaniac lmaooo there is no dream uni in greece they’re all equally bad and the way you get into them is by just taking a final test and scoring the required amount but so if you want study math at athens you need 13.5/20 and if you want to study at say an island they require an outstanding 3/20 and not because the test is hard you can search up the math section if u want but because math keeps getting reduced in public schools in fact I would not be surprised if my kids only learned basic algebra. So idk you can tell me your thoughts it’s not like I need to decide right now also sry for trash English :D
To clarify how bad the situation is, some math professor wrote in a greek forum that they had students who got into math but they didn’t know how to expand (a+b)^2 . It’s much much better at engi school most people who score 17.5/20 or more go there I know it’s pretty weird and also a few participate in undergrad competitions maybe as many as the actual math and physics department XD
Well, if the situation is that bad, have you considered anything outside of Greece? Maybe there can be scholarships to help you study elsewhere?
cant wait for the next video, best maths channel!
Thanks so much for the appreciation!
I am happy and "angry"!
I have the very bad habit of trying to solve the problem before watching the video. I've got at the same terrible integral which not even WolframAlpha wanted to solve.
So I've tried to solve it numerically and, before get the answer, I've got exactly the same problems of yours.
Finally, I'm very happy for this challenging exercise and for these great two videos you've made explaining everything!
Great work!! Thanks!
When I first thought of this question, I did exactly the same things, which is why I mentioned them in the videos! You are not alone :)
6:25 iam lost here ,someone please clarify me how it is 4dimensional domain,and we took square as analogy
Very good attempt!!! Regards.
Thanks!
So uhhh I would be lying if I told you I didn't come from the dream video lol. I did subscribe though for a reason related to this question: are your videos heavily inspired by 3Blue1Brown? I've been watching 3B1B for sometime now and I can honestly see resemblance in style after watching some of your past videos.
Awesome work by the way :)
Thanks! The style is inspired by 3b1b, but I don't use his python library to produce the videos.
@@mathemaniac wait so you made an animation library yourself akin to his?
Awesome video
Thanks!
I wasn't smart enough to figure this out, but I was smart enough to figure out the average distance for a random point in the circle from the center of the circle. I recommend everyone try to figure THAT out for yourselves. Then I tried to figure out the average distance from a random point in the circle to the point at the top of the circle. That one stumped me. Maybe you can do better. I was hoping that they would both result in the same value which would suggest (but not prove) that the answer was the same no matter what you picked as the first point. Here's an analogy that might help you sense what the right answer is. If the circle was a completely flat island in the ocean, 1 mile in diameter, and you were on the island, and a lightning storm was approaching, where would you lie down on the island to be as FAR AWAY as possible from the upcoming randomly-located lightning bolt? (Since even PROXIMITY to a bolt can kill you, not just a direct hit.) Assume that lightning doesn't strike water. Would you stand in the middle or the edge of the island? Would it matter? The edge of the island is obviously further away from most land on the island than the middle of the island. So the edge would be the safest place (since proximity matters.) This means that the average distance between two random spots on the island does depend upon (and vary based on) the location of the first spot. That means that the average distance of a bolt from the center of the island, which is easy to solve, isn't the correct answer for the original problem.
at 0:59 I went back to watching scary pranks, but I will be back
oh, this looks interesting,... quadruple integral nope nope nope... monte carlo it is then
great videos. came from speedrunning to dream to here
Thanks so much for staying haha :) We will solve the problem analytically afterwards, using some other approach, but we will first take a look at the numerical simulations first!
Maybe avoid the random issues and just use a uniform grid of points. As the number of points increases so does the accuracy of the answer.
In 4 months this channel gained like 20k subscribers lol
U the Boss !
If there's part 2 coming out?
Do you have a live online math class? I would love to be one of your students.
No, not really...
nice
I'm not sure you know this but you can use `` to start the quotation marks instead of '' which prints the closing quotation. Anyways, nice video!
Thanks! I know that in the middle of my most recent Dream video haha:)
@@mathemaniac yeah :)
this seems at first intuition, that it would tend towards 0 (like an asymptote or maybe a 2d Gabriel's Horn). Then I realized that these are 2 random points and not the permutations of every point
Come on with the part 2 already!!!
Haha :) My videos take a long time to produce, and I am not doing UA-cam full time, so sorry that you might have to wait a bit for part 2.
Oh that was great math joke.
My instinct was 0.5 diameter (radius). Well, turned out to be a very bad estimation.
Not TOO bad - it is just 10% away from the answer haha :)
I think you have been inspired by 3B1B............nice work.......
Thanks for the appreciation! The style is indeed inspired by him :)
@@mathemaniac I would like to do similar thing in my native language in future....... Because there is no such good math related material available in hindi Or marathi language.....
Is there a solution of the "horrible integral" ?
Watch the remaining parts of the series to find out!
very cool
Thanks!
@@mathemaniac
oh wow i didnt think you would actually reply, thank you :D
Time ago random was replaced by frecuency. Its more simple and academical to apply and study. I assume u mean same probability with uniform distribution on R^2
That's exactly what I try to say in the video!
My guess would be pi. The circumference divided by the diameter.
I could never have imagined I would laugh so hard at an irrational fraction punchline.
As you should!
A straight line
*”Random”*
Nice Swedish quotation marks.
Made a rookie mistake there in LaTeX haha, should have used `` instead of ".
4 dimensional domain at 6:48?
Yes, to specify a point in that domain, you need to specify two different points, and both points live in a 2-dimensional domain, so altogether you need to have 4 dimensions. More simply, to specify a pair of points, you need 4 pieces of information: x and y coordinates of both points.
Thank you 💐
My lazy solution
NIntegrate[4 r1 r2/(2 Pi) Sqrt[(r1 -
r2 Cos[\[Phi]])^2 + (r2 Sin[\[Phi]])^2], {\[Phi], 0,
2 Pi}, {r1, 0, 1}, {r2, 0, 1}] = 0.905415
I'm curious to see how you get the analytical result.
Monte Carlo solution:
local reps = 1e7
local x1, y1, x2, y2, avg
local tot = 0
for idx = 1, reps do
repeat
x1 = math.random() * 2 - 1
y1 = math.random() * 2 - 1
until x1 ^ 2 + y1 ^ 2
Gonk
The limiting factor to the degree of accuracy is the resolution or limit to the number of points and the minimum distance between them.
You invoke random points, but the original questions asks for the average distance between any two points within the disc(circle). Nothing about that requires "random". Choosing two points at random serves no apparent purpose. Going through every possible combination of points would yield the most accurate answer, and going through them randomly or sequentially should not matter.
If there are an infinite number of points with and infinitely small minimal distance at which points could be in proximity to each other, then the average approaches the radius of the circle. However, limiting the possible number of points and therefore the minimum possible distance between the points the average distance between the points would tend to be greater than or equal to the radius of the circle if implementing true equal distribution. Selecting random points and or limiting the number of points will bias this and affords the exception that the average could be greater or less than the radius.
Again, I consider calculation approaching an infinite pair of points to derive an approximation of the average being the radius by factoring in a minimum distance of 0 and a maximum distance equal to the diameter.
If you just want to pick two points at random, then the answer cannot be precise and is affected by probability. The average distance has the greatest probability to be the radius of the cirlce, when approaching an infinite pair of points.
If limiting the number of possible points, there are more points that exist the farther you get from the center of the circle. Consider area calculation. A circle of radius 1 has an area of pi(1^2)= pi(~3.14159) if we double radius to 2, the area grows by a factor of 4. pi(2^2)=4pi. If we double the radius again to 4, pi(4^2)= 16pi. Just demonstrating area quadruples with doubling the radius. The number of points that exist, if evenly distributed, increases proportional to area. If we compare the number of points in a 0.5 radius of the center in relation to area, we get pi/4. Then we can look at the area of the next 0.5 which would be pi. The area of the circle with radius 1, discluding the area of the circle with radius 0.5, is pi-(pi/4) or (3/4)pi. This means that only 25% of the points should exist within the first 0.5 radius circle. 75% exist between radius 0.5 an 1.
What a lovely rabit hole to chase down!
Did this in code, got about ~110 with r=200
Do you mean the disc being 200 in radius, and you get 110 as the average distance? That's quite far away from the analytic value I said in the beginning of the video... We will talk about how to numerically simulate properly in the next video though :)
You cannot select the random point out of real numbers from let say 0:1 range. The problem, it will take an infinite amount of time just to select one number. The same problem is with surface of circle. math.stackexchange.com/questions/1587303/picking-two-random-real-numbers-between-0-and-1-why-isnt-the-probability-that
Just open the video, if it ain't 0.5 I'll die
RIP
Two words to scare people: measure theory
Poggg
So, as I understand it, it's because magic.
It really isn't - will finally tackle this question in the next video.
to everyone: watch it at 480 p otherwise you waste bandwidth. 60fps is completely unnecessary for these kind of videos
I'm First 🥇
NO, I left a comment 13 hours ago, so you are second! Just kidding haha
@@mathemaniac lol
algorithm comment
algorithm reply
algorithm comment engagement
Your quotes are wrong. You’re using two right quotes, you need to use a left quote and a right quote.
Yep - I know how to use the quotes in LaTeX now in my later videos, but I couldn't edit it directly on UA-cam.
Mid roll ad spam in an 8 minute video????? Unsubscribed
First you make a big mistake, then you long explain why were you wrong... not impressed...