Updated probability density function video

To add to the previous comment: I really liked the first two parts of the Probabilities of probabilities series :) Can you share where you stand on a possible part 3 sequel – is it something you're still thinking about, are there any blockers that prevent you from getting to it, or is it a path that you have shelved for the forseeable future (or maybe even permanently)? Thank you so much for making these videos and for taking the trouble to keep us posted!

Dude where is part 3? This is perhaps the most interesting video you have made. The reason I just subscribed to your Patreon. It's been 6 months :(. Please finish the series.

This series has been awesome so far! and shockingly timely for work reasons I can't get into. I'm not sure if it is outside the scope of this channel (I think by now you've proven that's not possible) but could you take a stab at Design of Experiments? It feels deeply related to this series, from what little I understand of it. I am a physicist/engineer, not a statistician, so I have never intuitively understood DOE. I can obviously follow directions of a statistician, but it would be better if I had an intuitive feel for it. Plus, it feels like one of those topics that will have powerful, wide reaching implications, which are my favorite in math.

Aaron Rips

Totally right. I made a brief mention of how the rules are the same for "countably infinite", but you're spot on that saying discrete would better capture it.

3blue1brown

From 5:58 on you talk about finite settings vs continuous ones. I feel that 'finite' may not be the best word here. Finite is sometimes synonymous with limited. However, the continuous contexts are also finite (limited) in a sense that x can only take values between [0 ,1]. A better word may be 'discrete'. Also, one could have an infinite setting but a probability >0 for each value of x. For instance, the probability mass function such as P(x = i) = 1/(2^i) for i = 1, 2, 3, ... assigns probability >0 to each x even tough x goes to infinity and P(x \in [1, 2, 3, ...]) = P(x = 1) + P(x = 2) + P(x = 3) + .... = 1/2 + 1/4 + 1/8 + ... = 1. I guess this is just a more general point, but it could be helpful to differentiate between infinite discrete settings and infinite continuous settings.

thanks for the nice videos. in which order would you recommend i watch the probability videos and is the beta distribution relevant as the previous video suggested or is it an upcoming video?

It's a very small detail, but around 7:12, the arrow on the right hand side goes a little bit out of the screen.

Nice! I wish I had seen this when I was a student. My teacher shows a dartsboard and an arrow. Teacher: "When the needle is a point, the measure of hitting probability is always zero, even if you hit the target." Student: "But you hit somewhere on the board, right?" T: "But, the measure of the probability of hitting a point is zero." S: "Everywhere on the board?" T: "Yes. Because the area of the needle's head is zero..." I think we had no attention that he told us "measure" every time. It took a while to realize that the "measure" is an important word. Thanks a lot.

Hitoshi Yamauchi

You're totally right that the intuition of replacing sums with integrals as you go from discrete to continuous contexts is generally good for students to learn. But honestly, I was always a little unsatisfied with this when I first learned it all, especially in the context of probability. It felt more like pattern matching than like getting to the bottom of what was really going on. In this case, the sum-to-integral swap didn't explain the paradox that the probability of any specific value is 0, yet all together they make 1. There might be some vague reference to "infinitesimal" probabilities, which you could perhaps make rigorous with some nonstandard analysis, but I felt like I only came to peace with it all when it became clear that the underlying rules were different from what I thought they were.

3blue1brown

Thanks for the catch!

3blue1brown

"I've reworked the video a bit". I'll say!

Instead of jumping to measure theory, when you consider PDF, why don't you just state that "you just need to replace the sum symbol by the integral symbol; and add a `dx`. This works because the integral is only the limit of the sum of surface of the rectangles, when they width go to 0. By the way, those notions where already considered in the essence of calculus series" Clearly, you would not be able to consider the case where 0 has a 50% probability anymore, and so it makes sens to mention that more complex theories are required in more complex cases. But I see no reason to directly mention measure theory when you could mention notions seen in your other video and basic calculus courses

arthur milchior

Really cool! Maybe it would be nice to connect more to the overall problem of updating distributions. E.g. maybe show how the probability of being on a given interval increases as the distribution becomes tighter.

same for the "ranges" comment at the top around 6:53

Edith Dubiner

Hi, awesome as usual, but around 7:28 it might be nice to reconnect to the previous blocks and say that the probabilities in such a case are still asked about intevals and not points and then say that the issue is handled in Measure Theory.

Bring back the Nikola factory :)

Hi Grant, I'm a huge fan of you, and I'm studying graduate statistics now. The introduce of measure theory in this video is more than awesome. Do you have any recommendations of books or videos about measure theory with an intuitive writing style instead of being purely proof-based?

At ~1:56, the popup that says "Proving this statement is actually a fun puzzle, see the description" - is on the screen for too short a time - I had to go back a few times to catch it. Otherwise, awesome so far! :D

Hm.. I sort of feel like you're piling on too many philosophical worries at once. When I teach probability I bring up the pdf in one context without worrying about Bayesian analysis, so that when we do get to Bayesian statistics we can focus on the strangeness of asking about the probability of a probability and not have to think about what continuous random variables mean

Kevin Iga

'Beginner' is spelled wrong in the title card around 1:02.

Frank Wales