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

Подписаться 6 млн

Просмотров 645 тыс.

50% 1

Adding random variables, with connections to the central limit theorem.
Help fund future projects: / 3blue1brown
An equally valuable form of support is to simply share the videos.
0:00 - Intro quiz
2:24 - Discrete case, diagonal slices
6:49 - Discrete case, flip-and-slide
8:41 - The discrete formula
10:58 - Continuous case, flip-and-slide
15:53 - Example with uniform distributions
18:42 - Central limit theorem
20:50 - Continuous case, diagonal slices
25:26 - Returning to the intro quiz
Thanks to these viewers for their contributions to translations
Hebrew: @DavidBar-On, David Bar-On, Omer Tuchfeld
Spanish: Derek Lacayo
------------------
These animations are largely made using a custom python library, manim. See the FAQ comments here:
www.3blue1brown.com/faq#manim
github.com/3b1b/manim
github.com/ManimCommunity/manim/
You can find code for specific videos and projects here:
github.com/3b1b/videos/
Music by Vincent Rubinetti.
www.vincentrubinetti.com/
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with RU-vid, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
Various social media stuffs:
Website: www.3blue1brown.com
Twitter: / 3blue1brown
Reddit: / 3blue1brown
Instagram: / 3blue1brown
Patreon: / 3blue1brown
Facebook: / 3blue1brown

Опубликовано:

9 июн 2024

Ссылка:

Скачать:

Готовим ссылку...

Добавить в:

Мой плейлист

Посмотреть позже

Комментарии : 763

@3blue1brown 11 месяцев назад

Next video: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-d_qvLDhkg00.html

@adityakumar2803 11 месяцев назад

❤

@mesauce 11 месяцев назад

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

@Eta_Carinae__ 11 месяцев назад

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 11 месяцев назад

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

@pa.l.2499 11 месяцев назад

@@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 11 месяцев назад

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 11 месяцев назад

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 11 месяцев назад

@@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 11 месяцев назад

@@TengzhichongYou made me laugh ... Thanks

@UnknownCleric2420 11 месяцев назад

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 am 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.

@3blue1brown 11 месяцев назад

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

@tparadox88 11 месяцев назад

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 11 месяцев назад

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

@Dinnye01 11 месяцев назад

@@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 11 месяцев назад

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.

@Inspirator_AG112 11 месяцев назад

*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 11 месяцев назад

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

@Inspirator_AG112 11 месяцев назад

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

@TheTKPizza 11 месяцев назад

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 11 месяцев назад

Nice

@apnatime4831 11 месяцев назад

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

@glennpearson9348 11 месяцев назад

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 11 месяцев назад

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 11 месяцев назад

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

@debrachambers1304 11 месяцев назад

That sounds pretty convoluted.

@akilvarmantikvar 11 месяцев назад

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 11 месяцев назад

@@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.

@her0blast 11 месяцев назад

Babe wake up, funny math guy just uploaded

@blackholesun4942 11 месяцев назад

Funny?

@yarlodek5842 11 месяцев назад

“I like your funny words, math man”

@ripmorld9909 11 месяцев назад

Cute pie creature !

@Hecarim420 11 месяцев назад

Yay, new whity math 👀ツ

@Tepalus 11 месяцев назад

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

@petergilliam4005 11 месяцев назад

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

@FiliusPluviae 11 месяцев назад

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

@vigilantcosmicpenguin8721 11 месяцев назад

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

@dangoyette 11 месяцев назад

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.

@ReyhanMehta 11 месяцев назад

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. ❤

@rahulsingh7508 11 месяцев назад

Very few RU-vidrs 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 11 месяцев назад

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

@siddharthnemani5301 11 месяцев назад

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

@Zach010ROBLOX 11 месяцев назад

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

@davidgillies620 11 месяцев назад

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).

@vidblogger12 11 месяцев назад

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!

@domenicobianchi8 11 месяцев назад

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 11 месяцев назад

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

@0utOfSkill 11 месяцев назад

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.

@rmyers99 11 месяцев назад

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!

@amos9274 7 месяцев назад

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! ❤

@tka4nik 11 месяцев назад

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 :)

@bentationfunkiloglio 11 месяцев назад

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

@Alfetto8 11 месяцев назад

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.

@vesk4000 11 месяцев назад

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

@WobblesandBean 11 месяцев назад

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

@vesk4000 11 месяцев назад

@@WobblesandBean Thank you!

@spideybot 11 месяцев назад

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

@fabiontona 11 месяцев назад

Good luck!❤

@laural4976 11 месяцев назад

Finally the probability series we waited for :)

@riverland0072 11 месяцев назад

Exactly! and he started it without letting us know

@ScottPenick 11 месяцев назад

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 11 месяцев назад

Same!

@haileycollet4147 11 месяцев назад

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)

@colin8923 11 месяцев назад

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.

@DrPillePalle 11 месяцев назад

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

@cassandrasinclair8722 11 месяцев назад

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!

@whitewalker608 11 месяцев назад

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!

@Atlas_Enderium 11 месяцев назад

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

@guyedwards22 11 месяцев назад

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

@leflavius_nl5370 11 месяцев назад

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.

@hiennguyenphuong739 11 месяцев назад

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! 🤩🤩🤩

@mikealexander7017 11 месяцев назад

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!

@micahbergen3791 11 месяцев назад

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 11 месяцев назад

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 11 месяцев назад

Your students are lucky to have you as a teacher!

@jacksonstenger 11 месяцев назад

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

@apnatime4831 11 месяцев назад

@@jacksonstenger k DUDE chill 😎 🤙 🤘

@user-ww5tz4iu5p 11 месяцев назад

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!

@GabrieleCannata 11 месяцев назад

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

@Sky-pg6xy 11 месяцев назад

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 11 месяцев назад

Well said.

@stratfanstl 11 месяцев назад

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.

@philippus1807 11 месяцев назад

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 11 месяцев назад

Thanks!

@mitromanzukal9216 11 месяцев назад

Beautiful and wonderful video! Thank you for the clear explaination!

@Inspirator_AG112 11 месяцев назад

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 11 месяцев назад

Your mom could be geometrized

@avinashreji60 11 месяцев назад

@@idontwantahandlethoughwhat are you 12?

@gauravjagtap2620 11 месяцев назад

@@idontwantahandlethough new to internet boy ? Huh

@ronm3245 11 месяцев назад

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.

@Greg-McIver 11 месяцев назад

I find your videos absolutely amazing! Thank you for the time and effort. The moving graphics are so well done.

@marcobecchio527 11 месяцев назад

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

@justinbond1609 11 месяцев назад

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!

@cassandrasinclair8722 11 месяцев назад

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

@drgothmania 11 месяцев назад

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

@kylebowles9820 11 месяцев назад

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!

@giovannironchi5332 11 месяцев назад

Thank you for this one! Can't wait to see it when i finish working today!

@lucasg.5534 11 месяцев назад

You've got some serious cojones putting this out the day before my probability & statistics exam.

@JackDespero 11 месяцев назад

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

@jak4002 11 месяцев назад

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

@yongliangteh7957 11 месяцев назад

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

@jschlesinger2 11 месяцев назад

Your videos are calming and engaging. I never thought that math explainer videos could be calming ...only anxiety provoking or boring or both.

@mpalin11 11 месяцев назад

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

@jrioublanc 11 месяцев назад

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

@lauram9478 11 месяцев назад

❤ Thanks Grant! Nice to have you back!

@fightme5543 11 месяцев назад

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

@eveeeon341 11 месяцев назад

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.

@AmoghA 11 месяцев назад

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.

@dylanparker130 11 месяцев назад

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!

@prosimion 11 месяцев назад

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!!!!

@Dezdichado1000 11 месяцев назад

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.

@multiarray2320 11 месяцев назад

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.

@Elristan 11 месяцев назад

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

@BS-bd4xo 11 месяцев назад

Perfect timing! Just finished Probably Theory.

@Julian-tf8nj 4 месяца назад

amazing insight, superbly explained with your soothing voice - a great mix of enthusiasm with a calm energy!

@MrBabausse 11 месяцев назад

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.

@fenrisianwolf9229 11 месяцев назад

Beautiful, thank you! So relaxing and good quality content, thank you again! :)

@fierydino9402 11 месяцев назад

Thank you very much!!! So eager to see the next video!!!!

@tveleruusk 11 месяцев назад

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

@alexbaker3547 11 месяцев назад

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.

@pushkal8800 Месяц назад

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.

@Neural-Awakening 11 месяцев назад

Very informative and well described. Thank you very much for this!

@vivekdabholkar5965 11 месяцев назад

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

@maibster 11 месяцев назад

This is amazing! Such an overwhelming amount of profound realisations hit me while watching this. Thank you so much for your videos

@FeanorMorgoth 11 месяцев назад

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

@guillaumewuthrich2889 11 месяцев назад

The video's are always great but this one is awesomely awesome.

@peterboylan8560 11 месяцев назад

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

@mastershooter64 11 месяцев назад

Now let's multiply two random variables

@avi12 11 месяцев назад

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

11 месяцев назад

The visuals have reached a new level. Really well done.

@jameshughes6078 11 месяцев назад

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

@ebrombaugh 11 месяцев назад

Amazing synchronicity - I was just researching how to make non-uniform random number generators in an electronic music application. Did some experiments in Python and came up with a lot of the same graphs as those shown here.

@jogloran 11 месяцев назад

I get inordinately excited when I see a new video in this series.

@matejmazurek8392 11 месяцев назад

Incredibly wonderful video! Good job!

@mahadlodhi 11 месяцев назад

Great vid as always, undying gratitude for grant

@Impatient_Ape 8 месяцев назад

Brilliant!! Truly well-done Grant, bravo!

@Kiyotaka-ym1md 11 месяцев назад

Great video as always. Appreciate your work.

@BattleHerb 11 месяцев назад

The best part about going to harder and harder math classes is being able to rewatch your videos and know what on earth your going on about

@Call-me-Avi 11 месяцев назад

Thank you for all your work my man. ❤

@ominollo 11 месяцев назад

Beautiful explanation 😊

@sherifffruitfly 11 месяцев назад

Yay new episode day of my fav binge watch show!

@Toto-cm5ux 3 месяца назад

Super cool! I never thought like that! You explained to us the simply deep reason why the convolution is used!

@zacharywong483 11 месяцев назад

Very nice video, as always!

@_hollister9515 11 месяцев назад

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! 😊

@Ilya-iu5ih 11 месяцев назад

so beautiful, thank you!

@11amanie 9 месяцев назад

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☺️

@versacebroccoli7238 11 месяцев назад

This channel is always fire but I am so hooked in since the convolutions video!

@ammardian 11 месяцев назад

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 6 дней назад

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

@superman39756 11 месяцев назад

Awesome video! Keep going w probability and statistics please. There is so much more to cover and seeing these concepts visually explained is extremely helpful!

@superman39756 11 месяцев назад

Please cover stochastic calculus and SDEs at some point 🙏 some concepts come up everywhere and should be well suited for visual explanations.