Summer of Math Exposition

Very easy to follow. Good!

Very nice! Thinking of evaluating a random variable as a matter of picking a grain of sand (or tiny bead) from under a pdf and finding its x-coordinate is very straightforward and intuitive, and the physical demonstration at the end really brings the point home beautifully. I imagine that students would respond very well to this way of thinking, and to this video in general.

I find that too many SoME entries rely on flashy hi-tech “smokescreen” in their videos, which looks great but often feels like a substitute for thoughtful pedagogy. Not this one. Well done!

Loved the physical demonstration of sampling! It really helps to build intuition.

Very very fine for introducing pdfs ! It would have been useful and illustrative to repeat the experiment at the end a large number of times to see how the empirical distribution approaches the “theoretical” one.

Very nice idea also to start with the quite mysterious images of the experiment. Very good teasing.

No

Great Proof & Example! Nice execution, with both accessible explanations for those who don’t know as much in terms of the equations, but with plenty of equations for those who want to follow along that way.

I loved the intuition and practical effects

Good video. I liked how the explanations about probability density functions were intuitive and made sense when explained using area under graph. The video also seems concise and clear.

The ending part could have used some more explanations. Picking the random variable by choosing X coord using the beads experiment didn’t tie into the bigger picture very smoothly. Also, the video glides over some basic concepts pretty fast - so it might not be very beginner friendly.

I think the sand example is an interesting new way to explain pdfs, and I really appreciate the effort to actually construct the physical example. If I had to give one complaint, I think that mentioning the distinction between discrete and continuous random variables could do a lot to clarify the issue for first-time learners of the subject - for instance, understanding that continuous variables don’t accumulate probability at specific points does a lot to motivate the relevance of “area under the curve”

The idea with beads is excellent! It is a very visual way of making the abstract concept of a probability distribution a bit more tangible.

The narration could be made more active, at the moment it is slightly monotonous.

I enjoyed the graphics, the presentation, and the experimental device. I was a little bit lost at the beginning because I didn’t know what was the goal of the video. Perhaps, stating the goal at the beginning would make the video more engaging for new audiences. I believe the goal was to give a physical interpretation of probability density functions, which are a fundamental tool in the study of random variables. Perhaps, including an application of PDFs would have helped… For example, one could think of a random variable as a generalization of a real number in which we allow for uncertainty. In the same way that we can transform a real number x by a function f to obtain a new number f(x), we can transform a random variable X through a function to get a random variable Y… In this case, the function f would transform the pdf of X into the pdf of Y. I wonder if there is a way of showing that using your device, similarly to a Galton board… In any case, I am just rambling, i enjoyed the video and i am looking forward to see more of them!

Very cool video and I liked that you had a device demonstrating how sampling is done. When I started watching the video I thought that it would go more into the direction of Inverse Transform Sampling, but I forgot that for that you need the CDF and not PDF. Either way very clear explanation with a good pacing.

I am quite impressed with this video. With how important probability density functions are, I’m sure that many viewers will be able to appreciate the inherent value of this explanation. There’s not much I can think of that would be confusing for the target audience. As expected, a little bit of calculus is involved, but a deep understanding of integrals isn’t super necessary; the video explains just enough so that everything makes sense. Visualizing the area under a probability density function is a cool idea for intuition, and I’m actually surprised that I haven’t been able to find anyone else use something like this. The entire video is structured around this key takeaway, and I’m sure that it’ll stick around in many students’ minds whenever they see a probability density function. Indeed, I can say that it’ll stick around in mine.

Now, with all of these positives stated, I admit that the video doesn’t reach the absolute best of the best. It has one cool idea that it presents well, but the rest is just a fairly standard lecture on probability density functions. Of course, it never aspired to anything greater, nor am I saying it should have; I’m just saying that by its nature, a lot of what it has to offer has already been well-covered, and it doesn’t reach mind-blowing status as a result.

I can also offer a few more specific criticisms. Firstly, a few inquiries can be made into the nature of these grains of sand. Are they “infinitely small”, for whatever meaning that has? If not, if they have a finite size, then what happens when your area gets too tight to fit any grains of sand in it? This would’ve been a good opportunity to use the concept of a limit: we simply use finer and finer sand, the size approaching but never reaching zero. As we do, our sampling gets better and better. Secondly, at the end, I would have liked to see the sampling repeated over and over again. This would let the viewer empirically see how the value of the probability density function relates to the probability of sampling a given number, just to make the concept as clear as possible. Finally, the video creator mentioned plans to construct similar devices at the end. I’m very much not making any demands here, because as I’ve established, I already think the video is good as-is; however, it remains the case that a version of the video which actually contained those similar devices would have been strictly better.

Overall, I consider this an entirely solid project. Score: 7.50.