Convolutions | Why X+Y in probability is a beautiful mess

[@3blue1brown]

Next video: kzbin.info/www/bejne/mpDUp396ndCaZpI

[@adityakumar2803]

❤

[@mesauce]

*Me watching this video having no idea what is happening but watches anyways*

[@Eta_Carinae__]

Hey Grant. Not to make a request, but I think it's a pretty neat video idea, being a relatively untapped vein of math communication: have you thought about doing a video on stochastic calculus and Itô processes?

[@multiarray2320]

now its the first time ive heard about this because i disabled community posts :/

[@pa.l.2499]

@@Eta_Carinae__ or even more off topic, yet. Your own take on visualizing fractional derivitaves with the Riemann-Liouville, or some other approach? While not apparently useful, a newer math topic like this always is fresh to see a video on. Is extending this idea into the complex domain or R^3 space possible as a visualization?

[@johnchessant3012]

I wonder how many non-math people never would've thought they'd find themselves on the edge of their seat waiting for the next video in a series on probability theory. Truly a beautiful animation and explanation of this topic!

[@MattRose30000]

As someone who hated stochastics in middle school and is now working with applied statistics and machine learning, I just wish these videos had existed sooner 😅 I've always been a fan of geometric intuitions, and this is why this channel does stand out so much to me. Grant has a talent of making abstract things graphical.

[@Tengzhichong]

​@@MattRose30000 seriously though. it all felt like chores when I was a child; the supervisor for the reinforcement learning on us kids could have tuned the model better :P

[@simonmasters3295]

​@@TengzhichongYou made me laugh ... Thanks

[@UnknownCleric2420]

Having just come out of a Calculus 1 class, I can look at these videos with a whole new world of understanding. Before, I had watched these videos because I thought it was cool and interesting to know what was possible with mathematics. But now that I have learned how to take a derivative and an integral, I can follow along with the processes much closer, and gain a better understanding of how these tools of calculus are applied to various problems in mathematics. It's much more fun this way, and makes me feel like the effort I put into the course meant something. Edit: Took more than entire year to realize we mistyped "an integral" as "am integral." Whoops.

[@3blue1brown]

Wonderful to hear. Calculus really does unlock a whole new world after you take it, including essentially all of physics

[@tparadox88]

Calc 1 was the first time I was excited to learn math for years. Derivatives and integrals feel less like a mechanical process and more like playing with numbers.

[@idontwantahandlethough]

Hell yeah! Isn't that such a wonderful feeling? 🤗

[@Dinnye01]

​​@@3blue1brownor me, calculus clicked in place when learning Physics I - and understanding the relation between velocity and acceleration. How the formulae I learned in High school are *derived* from each other. DERIVED. It was a WHOOOOAAAA moment. The word means more than face vakue. Everything just clicked. Your videos recreate that feeling. And I love it. I do grab pen and paper with your videos and calculate along. Best days!

[@nothayley]

I think this comment contains a really important point. I often see comments that are like, "wow this explained it so much better than my teacher" "why couldn't you just teach everyone" and things like that, but as flashy as these videos are and as simple as they present the concepts, you can't get full understanding of something in mathematics from just watching it. You have to actually do it, and practice it a lot.

[@glennpearson9348]

As a civil engineer by trade, the two convolutions I most enjoy are: 1. Convoluting a Unit Hydrograph with a Hyetograph to determine a given natural system's (or, "watershed") surface water conveyance response to a given rainfall event. Then, 2. Using multiple watershed responses (say, individual discharge points from streams), convoluting the intersection of multiple watersheds (streams) to determine a larger river systems response to various rainfall events. The Corps of Engineers has been using the concept of convolutions for decades to create flood probability maps for the entire United States. These maps, which establish the flood level for a given return-period storm, in turn, are used by insurance companies to determine the rate that should be charged for your flood insurance at your particular home. How's THAT for real-world application of convolution?!

[@pa.l.2499]

I bet wildlife conservation agents use this approach as well for reporting over-population for game based on crash report data. Like how many white tail deer are becoming a nuisance per convolution of crash statistics.

[@alejandrotenorio2327]

Also a civil engineer! What do you use to make these convolutions?

[@debrachambers1304]

That sounds pretty convoluted.

[@akilvarmantikvar]

As a teacher of actuarial science (insurance mathematics), I cannot wait to share this video with my students next time I teach about convolutions.

[@glennpearson9348]

@@alejandrotenorio2327 Several different ways, I suppose. The classic approach is that used by the old Fortran-based model, HEC-2 (later, HEC-RAS). However, there are other methods that found popularity after computational power increased. Two are the Runge-Kutta method and Taylor series expansion. These days, one can even apply Monte Carlo techniques to filter out some of the randomness of otherwise stochastic responses in complex hydrologic systems.

[@Inspirator_AG112]

*Side note:* I found a really cool method for geometrizing/visualizing geometric integrals. That is taking the function you want to integrate, graphing its square root in polar coordinates, and using the formula for the area inside of a polar graph; this becomes useful if the polar graph draws a conic section, which is actually not that hard to take the area of. *I have r/mathematics posts with examples (listed by title, from least recent to most recent):* • "Yesterday or so, I realized that polar graphs can be used to geometrize integrals..." • "I played around more with that cartesian substitution I discovered a month ago."

[@3blue1brown]

That's a really neat way to integrate squares of trig functions, I hadn't seen that before!

[@Inspirator_AG112]

@@3blue1brown: The solution for the integral of secant is also cool. It turns into the area of a hyperbola sector.

[@TheTKPizza]

Isn't this like basically a generalizaion of the famous solution for the Gaussian integral, where you transform it into 2D and then into polar coordinates? That is so nifty!

[@yudoball]

Nice

[@apnatime4831]

Bro I figured it out way before even for discontinuous functions .you take the langharian zeros of the function and put them in the gamma function . Basically this loops the area of function into a circle around origin. From where it's radius can be determined and using pi r square u find the integral. Also my post got 17.9 k upvotes

[@micahbergen3791]

I am the 7th-12th grade math teacher in a rural community, and I wanted to tell you that your videos have inspired me to learn Python so I can make interactive educational videos on topics and levels my students can enjoy. Thank you for continuing to deliver great content that inspires a love for math education.

[@apnatime4831]

No benifit bro ur rural children won't get any of that stuff just teach em the basics. Why waste money on those bastards only to be dissapinted

[@jacksonstenger]

Your students are lucky to have you as a teacher!

[@jacksonstenger]

@@apnatime4831Don’t criticize a good teacher putting forth extra effort. Actually, a teacher is probably what you need, to help you spell better

[@apnatime4831]

@@jacksonstenger k DUDE chill 😎 🤙 🤘

[@ReyhanMehta]

This is such perfect timing, Grant. I was just studying this from a textbook, and I wasn't able to gain an intuition on continuous convolutions; and here you are, to the rescue! Once again, we cannot thank you enough for your brilliant contribution to the world. Thank You, Grant. ❤

[@petergilliam4005]

Another priceless experience paired with a heartbreaking cliff hanging. Thank you for your work!!

[@FiliusPluviae]

I literally started gasping loudly and violently at the cliffhanger. Now can't wait a MINUTE for the next one...

[@vigilantcosmicpenguin8721]

This is the most cliffhanged I've felt from a 3B1B video. He's outdone himself.

[@leflavius_nl5370]

I begrudgingly took 6 months of Control classes for mechanical engineering, which is basically just lots of analog signal processing mathematics, and i don't think any of the subjects stuck. Demented unmotivated teachers didn't help, of course. Your videos have actually sparked an interest in this field for me, and made me understand stuff. Thanks man.

[@Zach010ROBLOX]

The diagonal addition representation instantly clicked as convolution, on a part that took me much longer to get when I first learned about conv. All your videos are made of these little moments and insights that are just so spectacular to visualize. Thank you

[@dangoyette]

I love the moments in his videos where he drops some profound truth (repeated convolution of any function produces a normal distribution), and I can only sitting there grinning in confused wonder at how that could be possible. It's kind of like getting to the end of a novel and reading the "twist ending" and that you never saw coming, but which fits perfectly.

[@vidblogger12]

I minored in statistics. I thought I understood everything I needed to know about the Central Limit Theorem. But that visualization with the repeated convolutions approaching a normal curve made it look like such an intuitive, obvious fact. I’d never looked at it that way before, and it was beautiful! Well done!

[@siddharthnemani5301]

Hey Grant. I know this isn't the right place, but I am really, really waiting for a course on statistics, just like your linear algebra one. The lectures will prove to be gems for me, especially in QM and engineering

[@tka4nik]

Coming from just finishing a Probability Theory course, these videos uncover a whole new world of visual understanding behind the formulas we've been using the whole semester, and its beyond enjoyable to shout "ITS CLT!" after the visualization, and be right :)

[@her0blast]

Babe wake up, funny math guy just uploaded

[@blackholesun4942]

Funny?

[@yarlodek5842]

“I like your funny words, math man”

[@ripmorld9909]

Cute pie creature !

[@Hecarim420]

Yay, new whity math 👀ツ

[@Tepalus]

Babe wake up! Someone just wrote a "Babe wake up!" comment!

[@amos9274]

Hey, I'm en EE student and just couldn't wrap my head around why a multiplication in the time domain equals a convolution in the frequency domain. With your shown approach of asking the question of what is the area of all the function products of the combination of arguments that equal x and the "sum trig identity" it suddenly is extremely obvious, tysm! ❤

[@0utOfSkill]

Man, I love how as I go through high school I understand each new video a little more, it felt like I understood this video fully and was always able to predict what came next. Great work, I really do appreciate you explaining these topics so incredibly well for free.

[@ScottPenick]

As a person with aphantasia, you'd think I'd be the inverse of the target audience here, but... I find these videos genuinely fascinating. They help me understand how other people conceptualize some of the same things I do, but with imagery instead of deductions from axioms. Great stuff.

[@jordanfarr3157]

Same!

[@haileycollet4147]

Agreed :) I had a similar thought when my aha! moment for this video was pausing on the Reimann sum text not anything visual, and had a bit of a laugh at myself (then pondered why I like the videos)

[@Atlas_Enderium]

I took my Signals and Systems course for my EE degree a year ago (which was basically just a math course on affine transformations, convolutions, and Fourier transforms on discrete and continuous signals/functions) and this was a nice refresher on the intuition behind convolutions

[@jak4002]

I'm an electrical engineering student and just finished learning FTs for system response stuff and this video has blown my mind to give me a deeper understanding of all the math I did all year. Thank you so much

[@davidgillies620]

Back in the days when mainframes had fairly fast processor-level pseudorandom number generators but relatively slow transcendental functions, a common way of getting a semi-decent Gaussian-distributed variable was just to sum three or four variates from the hardware RNG, suitably shifted and scaled. I've actually seen this in some FORTRAN code for a particle accelerator simulation (which was eventually rewritten in C++ and became PYTHIA).

[@eveeeon341]

Oh wow, I'm someone who doesn't usually chime with visual explanations, algebra tend to resonate better with my understanding. But I was fully engrossed in the visual, kind of ignoring the algebra, and I literally said out-loud "that's anti-derivation, it's integration" and then looked to the right of my screen to see an integral. Brilliant work, as always.

[@domenicobianchi8]

I love the topic choice. I love how you're dealing with it. I hate i have to wait weeks for the next episode, but i know it worth it for the quality. I just wish i discovered your channel five years from now, so i had already the full serie. Thanks Grant for what you are doing and providing it here

[@WAMTAT]

Yeah, but in 5 years Grant will still be making awesome videos that you'll have to wait for.

[@ammardian]

As someone that looked into convolutions in the past but never quite understood them, this video really solidified my understanding that I couldn't quite explain before. Before I just saw it as a daunting operation that could help me with Laplace Transforms. Now, I can see it more as a 'comparison' operator between two functions. It acts as, essentially, an operator analogous to the dot product for vectors, by comparing how much of both functions at a given point are 'similar', in the same way the direction of two vectors with respect to each other is compared in the dot product. Thinking on it now, I see it almost the same as the idea of the FTC, but the FTC definite integral compares a function to the width of the interval you are integrating on. This acts as a more generalised version of that definite integral (not literally, just for lack of better phrasing) and compares a function to another. Thanks 3B1B, for another cracking video that really makes me enjoy Mathematics more and more by the day.

[@ugestacoolie5998]

woah, when you said "comparison" operator of 2 functions compared to the dot product of 2 vectors, something really feels linked together to me, thank you

[@Alfetto8]

It's always so sweet to see the intuition you bring to these topics. The smooth way everything clicks together. Probability is integral part of my work (phd in financial econometrics) and when doing advanced stuff it's easy to forget the beauty hidden in the most simple things.

[@RolandWinkler-s4m]

I studied math in university. And probability theory was always my weakest subject. I could never intuitively place the math and its implications in my brain. In almost all other subjects, like calculus, measurement theory, algebra, etc.. I had a clear intuition. Not in probability theory. Its hard to build that intuition. And this series, of convolutions and probability theory is actually plugging the holes that my university education left me with. I would have been a much more successful on the subject when I studied it with your videos to give me a hand. Thank you, Grant. Also, notice how the colors are chosen to be visible for people with red/green viewing disabilities? I dont have that impairment but I notice it nonetheless. Great work!

[@DrPillePalle]

You're making the world a better place, one video at a time. Thank you so much!

[@colin8923]

Your videos are weirdly comforting to me. Even if I don't fully get them, I really enjoy watching. Also, you made me really like math, I've been self studying calculus after watching your series on it.

[@rmyers99]

I didn't take any math past Trig and these videos make total sense to me. Wish they had this video for me back in 1994!

[@alexbaker3547]

I'm graduating with my BSEE degree, and this would have been extremely helpful for a couple of classes. Very insightful for you electricals that haven't done linear systems, or want to focus in communications.

[@cassandrasinclair8722]

You have a tremendous ability to hint at what's to come! First identifying the equivalence with the diagonal and then figuring out where it comes from using the formula before you even presented felt incredible, thank you so much Grant for this experience!

[@stratfanstl]

The visuals in these videos deserve to be played on a big screen TV hanging in the Louve. I can't imagine any better use of today's computational power and programming / animation tools than producing these educational videos that not only lift the veil around mathematical mechanics but provide insight into the world around us -- exactly what math is supposed to do.

[@justinbond1609]

You've really outdone yourself. My signals and systems class years ago would've been so much more... accessible? with these videos as an aid. Glad current students are able to benefit!

[@guyedwards22]

Every video you release breaks my heart with a cliffhanger 😩 Your content is so good Grant, I never want the lessons to end.

[@pushkal8800]

My man, 3 blue 1 brown loves Fourier transforms so much, that his animation of the eye, his channel logo, is literally converting a function from time domain to frequency domain. What an amazing hidden gem, such a cool way to put Fourier transform animation into you logo. Amazing.

[@nizogos]

One of the things I found tricky for continous convolutions is finding the rage of the integration,which I can kinda wrap my head around way easier when I define f and g as the actual function times its indicator value of its domain.Then,under the integral the product of indicator functions ensures that the product of the functions is zeroed out were its supposed to be,shaping the ends of the integration along the way.Your remark about discrete impropable events being 0 in the sum helped me cement this idea. Overall a pretty insightful video as always!

[@hiennguyenphuong739]

I have nothing more to say than the pleasant to watch your videos. You make me, a sixth grader understand calculus, topology and a ocean of beautiful math. The world becomes a much better place with your videos sir. Great respect! 🤩🤩🤩

[@mastershooter64]

Now let's multiply two random variables

[@rahulsingh7508]

Very few KZbinrs make a 30 min-long Math and Science video that is more fun to watch than a 15-second-long Instagram reel. Hats off to all of you!

[@brightsideofmaths]

Having 30 minutes fun is always better than having just 15 seconds :D

[@mearnest91]

You’re my hero. I’m quitting my corporate career to start my own business teaching math and excel. You and StatQuest are my inspiration.

[@11amanie]

Having studied AI your whole channel sums up my study in an so much easier way. Our teachers over complicated stuff or didn’t even bother to explain the underlying mathematical theories of the machine learning algorithms. So thank you very much sir. I am going to watch every single video☺️

[@whitewalker608]

I just finished your Discrete convolutions video and Residuals FFT that you recommended in that video. Was looking for your video on continuous convolutions. This is impeccable timing! Thanks for this!

[@mpalin11]

This is seriously better than a proper university lecture on the topic. Thank you for this video.

[@milleniunrealjaron]

As a music lover, I applaud the juxtaposition of Vince's "Occlusion" with Rubinetti's "Heartbeat"; and as a math lover, I welcome the 3D visual for how to apply continuous convolutions to different normal distributions. More!

[@philippus1807]

Hey Grant, i really enjoy your videos. Your explanations from simple examples up to the general concepts are interesting and feel natural. The understanding growing in mind is so satisfying. With no destraction by strict mathematical definitions, i find it easy to follow. Also the amazing animations arent just nice to look at, they do a great job in supporting the intuitive understanding. You fill the gap of explanations, that are missing in my university courses. Thank you for your work, im looking forward to the follow-up video ✌

[@3blue1brown]

Thanks!

[@cmilkau]

The combined density function (two inputs) also generalizes to the case where the two variables are *not* independent. In that case they just can't be factored into this neat product of two single-input densities. The two marginal densities f, g can still be extracted by integration along the coordinate axes.

[@mikealexander7017]

I wish these visualisations had been available when I was struggling to get my head round stuff like this 35 years ago! I remember using a convolution integral to solve some Laplace Transform problem in electrical circuit analysis, but being annoyed that I didn't really understand how it worked!

[@AmoghA]

Hey Grant! I just took a course on probability and statistics this semester and this video is a great way to review and reinforce the intuitions I have on the course just before the finals. I would love for you to make a series on calculus of complex numbers, talk about analytic functions, countour integrals and stuff like that. Even though I finished the course on that topic, I would still love for a 3B1B video/series on it and many would be interested too! I also would like to mention that most of the intuitions I have in maths, be it calculus or probability, is because I have watched 3B1B. I have a decently strong idea of what is going on in class because sometimes I can connect what I saw here and what I learnt there. These videos are excellent for communicating maths and my friends and I just love it! Thank you for what you do.

[@Sky-pg6xy]

Yes! Your visual Linear Algebra series was transformative for me, and I get the feeling that a similar series on mathematical statistics will also be.

[@xyzct]

Well said.

[@rohitraghunathan]

I love this way of looking at convolution. We kinda rushed through it in a signal processing class every other course I took afterwards assumed that we knew what it was and how it worked. Took me a few years of internalizing it till the whole picture clicked for me. Thins would have helped me a lot in college

[@dylanparker130]

Ah, 14:00 - 16:00 was so good. The explanation of "Where's that y gone?" and the joy in seeing how adding together 2 graphs of fixed shape can result in something resembling a travelling wave(let). Come away feeling inspired!

[@MrBabausse]

Thanks a lot for this video ! It might be far-fetched, but I work a lot on audio synthesis these days (programing my own synthesizers) and while I use convolutions A LOT (for effects, mainly), I didn't quite understand how it worked until your video. I'll have to watch it three or four times again, and make more researches, but I feel like something "clicked" while looking at it. Awesome stuff, thanks a lot.

[@davidritt7474]

Another great video with stunning visualizations. This is so nitpicky it's almost not worth mentioning, but around the 22 minute mark the video invites the inference that the marginal densities of X and Y will look the same as the side profile of the joint density. It's pretty easy to construct a case where they'll depart drastically, especially if you throw out the assumption of independence. The marginal density is showing the result of integrating over one variable while holding the other constant, while, setting aside perspective, the side profile is going to show the maximum density of one variable for every value of the other. There's no guarantee these will look the same.

[@marcobecchio527]

Everytime you make may 50 years old engineer mind explode with yourt wonderful videos! Thanks !!

[@lollo_gabe]

I am currently attending the first year of physics at uni and tomorrow I'll have to do the oral exam for my stats course. The professor explained continuous convolutions on the last lecture and this video just dropped... I think I'll carefully watch the video and be more thankful to Grant than I've ever been.

[@abirsadhu5538]

best of luck for your exam ❤

[@lollo_gabe]

@@abirsadhu5538 thx

[@jameshughes6078]

"an attractive fixed point in the space of all functions" Wooahhhh, that was a great insight/way of framing it

[@vesk4000]

Perfect timing, just 2 days before my Probability Theory & Statistics final at uni!

[@WobblesandBean]

I took probability last semester, this would have helped lol. Good luck on the final!

[@vesk4000]

@@WobblesandBean Thank you!

[@spideybot]

Good luck on the exam; may the nerd force be with you!

[@fabiontona]

Good luck!❤

[@corellonable]

Its probably no surprise to you but i think you should know that the videos you do and have put out throughout the years immensely help those of us who are currently or about to undergo a mathematical heavy education. In my case i am in Area Studies (middle east & north africa) but will be leaning into economics and hence these maths videos are insanely helpful to understand maths and statistics better. your content is super inspirational and im very happy to be here to witness it, thank you so much

[@PaulSteMarie]

The nice thing about the diagonal representation is that you can rotate the 3D graph so you're looking along the slices instead of through them. Then you can mentally squish the slices and form a 2d graph, sort of like the trash compactor in Start Wars. Each slice gets squished into a vertical line segment.

[@bentationfunkiloglio]

Great video. Wish your videos existed when I took stochastic processes!

[@Dezdichado1000]

Probability is really mind-blowing. There are rough analogues of CLT's that result in a distribution that is not normal i.e., The Tracy-Widom distribution, Wigner's semicircle distribution etc.

[@GabrieleCannata]

It took me 51 years, and a KZbin video from one of the best, but I finally got convolution. And the explanation was not convoluted at all!

[@_hollister9515]

I am working on a special distance defined as the similarity of 2 probability distributions, and one way to speed up the computation is to get the sliced version of that distance. This vid explains that idea behind pretty well! Thx! 😊

[@prosimion]

I haven't even started watching yet, but dude your awesome. I literally needed to learn the premise of the refined version of this in base 10. thank you!!!!

[@rammerstheman]

Really nice video, feel like this is the best I've followed along haha. I'm quite sure the convolution of these two Normal distribution will be another normal distribution. Otherwise the central limit theorem wouldn't make sense - every time you take the convolution, you get closer to a normal... But if you're already at a Normal and you take a convolution with another normal and it gets less normal... That wouldn't make sense. No idea what the standard deviation would be!

[@drgothmania]

Every time I learn about convolution, some amazing new thing surprises me. Thanks a lot.

[@vivekdabholkar5965]

You are awesome teacher! I have a Ph.D and I still enjoy the content and benefit from it due to deeper understanding.

[@mckinleypaul6943]

14:00 I think a good way to write the convolution that shows you the inherent symmetry between (f*g)(s) and (g*f)(s) is to use the dirac delta function to constrain the inputs such that x1+x2=s if f=f(x1), g=g(x2), for example: (f*g)(s)=\int \int f(x1)g(x2)\delta(s-[x1+x2])dx1dx2

[@avi12]

This video is beautifully made. I'm a university student and one of the courses this semester was a statistic course. This video was uploaded a few days before the final exam, a great way to sum up what I've learned in the past 3 months

[@AngieTheCatGD]

I like to watch these videos while doing chores in the background. As in, the chores are in the background. This is my main focus. Until I realise that it’s been over 4 hours and I still haven’t finished folding one basket of laundry. Like right now.

[@christopherli7463]

At 6:38 you say you are just having fun with the animations, and it does look really fun. But I'd like to add that it gives a deeper visual understanding too. And in some ways it is an immediate explanation for why the convolution has a weird length (like an odd number). So I'd like to say thanks for the 3D animations and please keep on making them since they give a real intuitive understanding of what is going on, collapsing on the diagonal.

[@rizalpurnawan23]

12:16 - "As a general rule of thumb, anytime that you see a sum in the discrete case, you would use an integral in the continuous case." This notion can formally be explained using measure theory. The key is the concept of Lebesgue integral, which is the generalization of concepts such as discrete sum and Riemann integral. As in the case of discrete sum, the underlying set is equipped with the counting measure. While in the case of Riemann integral, the underlying set is equipped with Lebesgue measure. And both the discrete sum and the Riemann integral can be generalized into a Lebesgue integral on a measure space. When I first realized this, no wonder why probability theory is full of measure theory. Just, wow. Anyway, it's a great video @3Blue1Brown! For the first time, I get to realize why on earth convolution is so important, and this is one of the whys.

[@Fenifiks]

Love the video, high school and college would have been so much more fun if the teachers would have used KZbin videos to explain the underlying math and physics!

[@Filup]

I took Fourier Analysis last semester and this would have been so nice to know. While I know nothing about the probability materials, the relationship to the functional analysis and measure theory is screaming at me!

[@fotnite_]

Oh don't worry, Fourier analysis is pretty relevant in probability. In fact, in general random variables are added together by using their characteristic functions rather than going through convolution, and the characteristic function is just the Fourier transform of the PDF. I wouldn't be surprised if he brings it up next video, since characteristic functions are very important to the proof of the Central Limut Theorem.

[@Filup]

@@fotnite_ Yeah, I think I have seen that from skimming a paper or two. We used the characteristic function of an interval (a pulse function), which has the Fourier transform sinc(x) (the normalised sinc function), which makes sense, since probability is all about Gaussian distributions. Perhaps one day I will dive into probability and statistics.

[@JackDespero]

You always show me new ways of thinking about tools that I have used for years. Thank you.

[@LimitedWard]

Robotics, computer vision, machine learning, computer graphics. All of these disciplines rely heavily on the concept of convolution in both the discrete and continuous form. So if anyone ever asks you "when would I ever use this in real life?" The answer is "any time you want to earn at least 6 figures".

[@jrioublanc]

Really educative way to introduce the convolution. I loved this video, thanks.

[@LovcraftianHorror]

I am currently reading 'Statistics for Experimenters' (Box, Hunter, Hunter), and just read the section on this. Your video is a really nice visual and accessible rendition of the content.

[@TheTrevorS1]

Where were these videos when I did my undergrad! I hope this elegancy and beauty inspires more students to continue.

[@decreasing_entropy3003]

I just took more than 1.5 hours to 'somewhat' understand a 27 minute video, and at the end of it, I can say that I understand 1% about convolutions. It has been a while since I have watched a complicated math video and simultaneously understood everything that has been said, but in this case, I did understand almost everything but for 3 things. This video is on the level of being a research paper in itself, it's so well made. The animation, the code that went in, the script and the approach to not bothers the viewers with pesky integrals, are as always, a 3B1B signature at this point. But I really hate cliffhangars, so I am already awaiting the next video in this series.

[@yongliangteh7957]

The convolution has been de-convoluted by this beautiful intuition.

[@Inspirator_AG112]

I have been exploring math on my own in the past month, and I have realized how many things could be geometrized. (Kind of a side-note)

[@idontwantahandlethough]

Your mom could be geometrized

[@avinashreji60]

@@idontwantahandlethoughwhat are you 12?

[@gauravjagtap2620]

​@@idontwantahandlethough new to internet boy ? Huh

[@ronm3245]

I'm 60 and I thought it was funny. Your mom is probably 12. Anyway, Inspirator's comment reminded me of how, to the Ancient Greeks, numbers were geometrical objects.

[@DrLogical987]

4:39 thanks for saying this. I think KZbin should require a warning for all CLT/IID videos that as mathematicalily satisfying as they are; very often the real world isn't like that.

[@peterboylan8560]

Already before this video, I know it's going to be amazing. Thank you for sharing your gift of teaching with us and I can't wait to learn today

[@fightme5543]

I'm genuinely in love with this video. I got obsessed with Monte Carlo simulation a while back and this is amazingly useful!

[@laural4976]

Finally the probability series we waited for :)

[@riverland0072]

Exactly! and he started it without letting us know

[@Elristan]

oh nonono I need the answer now! Truly beautiful and insightful, this video kinda revolutionized the normal distribution for me. Thanks!

[@konan4heather]

I appreciate this video, it helps fill in the gaps. I did find the step where the convolution was intriduced as an unnecessary step, forgive my narrow-mindedness. If the task is to find X+Y, I find it natural to immediately go to the most fundamental breakdown - "how do X and Y together distribute?" which is to analyze the joint distribution f(X,Y) , and then derive the aggregation (sum), f(X+Y). This doesn't help at all with the analytical derivation but it feels more natural - it doesn't require to know about convolution (which sounds like a lucky hack), and I can have the freedom of switching the aggregation function with whatever the objective may be. In the discrete examply - why worry about convolution if analyzing the joint table works just as well.

[@kylebowles9820]

Love this channel! Epic work on the math and the animations Grant! I'm studying path tracing in my little free time, this is all highly relevant!

[@alagaika8515]

Great video as always. Btw, you actually can write a symmetric definition of the convolution using the delta distribution and a double integral over f(x)g(y)delta(s-x-y).

[@cosi321]

This made me think about the whole thing in the Fourier Domain. Actually, the CLT says that no matter what function you have that is the Fourier transform of a pdf (smooth because of the tails and symmetric because the pdf is real), you get the Bell curve by stretching it out and multiplying it with itself. And this even makes sense as the symmetric Fourier transform of the pdf will look like some version of (1-x^2) very close around zero. And when you stretch that and multiply it by itself, i.e. (1 - x^2 / n)^n, you get exactly the limit formulation of exp(-x^2). So cool how it wraps around!

[@trapkat8213]

Great visualizations. It took me ages to become comfortable with convolution when I was in university in the 80s. No visualizations back then.

[@jeyasitharamj6938]

Indeed a great visualization. With this we can easily figure out how to fing probability distribution of functions of random variables too. Say 2X +3Y, or X*Y, etc, but the formula you will arrive will get complicated as the derivative of the function and integration path.

[@multiarray2320]

i have to admit that your videos are challenging to watch because i am not good at math, but the reason i watch every video are the beautiful anomations.

[@ProfessorDBehrman]

When I first learned about convolution I was told to "slide one graph along the other" but this trick never made much intuitive sense. Thank you so much for explaining convolution intuitively.

[@jurjenbos228]

I wish I had seen this when learning all this: more fun and easier to remember than calculus class. And it makes it really clear where the square root of 2 comes from.

[@tveleruusk]

It’s always great to see how you bring in geometry to generalise and make seemingly abstract concepts become intuitively obvious. Fantastic teaching technique!

[@erickappel4120]

Just one word. "Wow!" You take explanation quality to unseen (at least by me) levels! Thank you!

[@FeanorMorgoth]

The best math channel by far. You rekindled my passion for math, thank you for the amazing content!

[@cassandrasinclair8722]

This is one of my favourite videos so far! Thank you!