5 ms·

Square roots and maxima

121 points by surprisetalk 2y ago

ndsipa_pomu 2y ago
Matt Parker's video on Square Roots and Maxima: https://www.youtube.com/watch?v=ga9Qk38FaHM https://www.youtube.com/watch?v=ga9Qk38FaHM
- raegis 2y ago
  The original post verifies the fact (experimentally) using simulations, and includes example code. However, the guy in the video you referenced does a lot of talking around the problem, and includes a VPN advertisement in the middle. I never heard of Matt Parker before, and I'm not knocking his talent (he has 1.24 million subscribers!), but the only coherent part of the video is where he includes an explanation from another channel, 3blue1brown.
  - stouset 2y ago
    They target different audiences. 3B1B tends to aim at those who want to know more of the underlying math and develop good analytical thinking, Matt Parker often keeps things at a bit more approachable a level for those who aren’t as inclined.
  - magicalhippo 2y ago
    Matt Parker is an ex-math teacher that does stand-up math comedy shows[1], FWIW. [1]: https://standupmaths.com/ https://standupmaths.com/
  - ndsipa_pomu 2y ago
    He's a comedian and maths communicator (including author). He appears in quite a few Numberphile videos too. He seems to be friends with quite a few YouTube maths presenters such as Grant Peterson and Hannah Fry (she seems to have moved over to TV presenting now which is good - I think she'd be an excellent choice to do a James Burke style Connections series as she also has a dry wit).
  - boltzmann64 2y ago
    People waste so many hours watching these types of video where the presenter talks a lot around the problem (has lots of fluff, comedy skit and ads). I think the presenter should have used this opportunity to introduce random variables, independence, Tschebyscheff inequality and explained the "bizzare" puzzle. Or if you are going to do this with simulation, then a introduction to Monte Carlo methods, and why such simulations work and provide correct results would have been a better us of viewer's time. But videos like this just state a fact and then handwave around the fact without hitting the core idea. The viewer leaves with a false sense of understanding, and keep wondering about the "bizzare" fact, when it is nothing but good ol' introductory probability. YouTube math needs reform.
    - ndsipa_pomu 2y ago
      I think there's room for Matt Parker's more "fluffy" type of maths presentations as well as the more in depth Grant Sanderson's visualisations. Matt is on a mission to popularise maths and it's easy for anyone that gets inspired/intrigued by a topic to dig deeper. Rather than a reform, maybe there's room for more maths-based videos to fill the gap between recreational and more serious maths?
dahart 2y ago
Either I haven’t seen this before, or forgot it, but it’s surprising because I use the sum of independent uniform variables every once in a while — the sum of two vars is a tent function, the sum of three is a smooth piecewise quadratic lump, and the sum of many tends toward a normal distribution. And the distribution is easy calculated as the convolution of the input box functions (uniform variables). Looking it up just now I learned the sum of uniform variables is called an Irwin-Hall distribution (aka uniform sum distribution). The min of two random vars has the opposite effect as the max does in this video. And now I’m curious - if we use the function definition of min/max — the nth root of the sum of the nth powers of the arguments — there is a continuum from min to sum to max, right? Are there useful applications of this generalized distribution? Does it already have a name?
- max_likelihood 2y ago
  Perhaps you are thinking of Order Statistics? https://en.wikipedia.org/wiki/Order_statistic https://en.wikipedia.org/wiki/Order_statistic
  - dahart 2y ago
    Ah, fascinating, I’ve never used Order Statistics. It doesn’t look exactly like what I was thinking, but there is also a continuum from min to median to max, similar to min to mean/sum to max. I’m not sure but I might guess that for the special case of a set of independent uniform variables, the median and the mean distributions are the same? Does this mean there’s a strong or conceptual connection between the Bates distribution and the Beta distribution? (Neither Wikipedia page mentions the other.) Maybe Order Statistics are more applicable & useful than what I imagined…
    - falseprofit 2y ago
      Median and mean are not the same distribution. Consider three uniform values. For the median to be small two need to be small, but the mean needs three. I think order statistics are more useful than what you described, because “min” and “max” are themselves quantiles and more conceptually similar to “median” than to “mean”. Trying to imagine how to bridge from min/max to mean, I guess you could take weighted averages with weights determined by order, but I can’t think of a canonical way to do that.
keithalewis 2y ago
Front page material? P(max{X_1, X_2} <= x) = P(X_1 <= x, X_2 <= x) = P(X_1 <= x) P(X_2 <= x) = xx. P(sqrt(X_3) <= x) = P(X_3 <= x^2) = x^2. It is late in the day when midgets cast long shadows.
- refulgentis 2y ago
  I felt the same way, came here to decide whether to comment something negative, saw your comment was just posted. But I first read the top comment from 2 hours ago, apparently this is news-ish to stats people because it's counterintuitive to common methods? shrug
prof-dr-ir 2y ago
If X1...Xn are independently uniformly distributed between 0 and 1 then: P(max(X1 ... Xn) < x) = P(X1 < x and X2 < x ... and Xn < x) = P(X1 < x) P(X2 < x) ... P(Xn < x) = x^n Also, P(X^{1/n} < x) = P(X < x^n) = x^n I guess I am just an old man yelling at clouds, but it seems so strange to me that one would bother checking this with a numerical simulation. Is this a common way to think about, or teach, mathematics to computer scientists?
- coliveira 2y ago
  > it seems so strange to me that one would bother checking this with a numerical simulation I believe that some people know programming but have little experience with mathematics, so the first thing they'll think about is to "check" numerically that something is true. Which in reality doesn't prove anything, so people should better spend the time to learn some math for these situations.
  - ValentinA23 2y ago
    "you can't learn maths on your own, you need a master" My math teacher during my second year in university, who also happened to be a chaos theorist working on cool stuff such as cryptography via chaos synchronization. He was by far the worst teacher I ever had in terms of mental calculation abilities, but he was also the more advanced. I remember a conversation where he explained how he would always implement his algorithms at least twice, on entirely different software and hardware stacks.
  - mistercow 2y ago
    I have, on many occasions, thought that I had an analytical solution to a problem pinned down, then checked it numerically and found that I'd made a mistake. It seems weird that there's any hostility towards a perfectly useful tool for checking your work.
    - monadINtop 2y ago
      Not the person you're replying to but I would assume their point was to highlight the distinguishing feature of mathematics - the fact that it's not only possible to completely prove something to be true, but in fact is the only thing you can actually do (yes I know about experimental math etc). But yeah that doesn't necessarily mean anything for pedagogy or just having fun and so on.
gxs 2y ago
Just a side comment on what a great little video. Short, to the point, and the illustrations/animations actually helped convey the message. Would be super cool if someone could recommend some social media account/channel with collections of similar quality videos (for any field).
- ajot 2y ago
  Well, that YouTube channel, 3Blue1Brown, is amazing. You'll see lots of great explanations with this kind of animations there.

121 points by surprisetalk 2y ago

Matt Parker's video on Square Roots and Maxima: https://www.youtube.com/watch?v=ga9Qk38FaHM https://www.youtube.com/watch?v=ga9Qk38FaHM

raegis 2y ago
The original post verifies the fact (experimentally) using simulations, and includes example code. However, the guy in the video you referenced does a lot of talking around the problem, and includes a VPN advertisement in the middle. I never heard of Matt Parker before, and I'm not knocking his talent (he has 1.24 million subscribers!), but the only coherent part of the video is where he includes an explanation from another channel, 3blue1brown.
- stouset 2y ago
  They target different audiences. 3B1B tends to aim at those who want to know more of the underlying math and develop good analytical thinking, Matt Parker often keeps things at a bit more approachable a level for those who aren’t as inclined.
- magicalhippo 2y ago
  Matt Parker is an ex-math teacher that does stand-up math comedy shows[1], FWIW. [1]: https://standupmaths.com/ https://standupmaths.com/
- ndsipa_pomu 2y ago
  He's a comedian and maths communicator (including author). He appears in quite a few Numberphile videos too. He seems to be friends with quite a few YouTube maths presenters such as Grant Peterson and Hannah Fry (she seems to have moved over to TV presenting now which is good - I think she'd be an excellent choice to do a James Burke style Connections series as she also has a dry wit).
- boltzmann64 2y ago
  People waste so many hours watching these types of video where the presenter talks a lot around the problem (has lots of fluff, comedy skit and ads). I think the presenter should have used this opportunity to introduce random variables, independence, Tschebyscheff inequality and explained the "bizzare" puzzle. Or if you are going to do this with simulation, then a introduction to Monte Carlo methods, and why such simulations work and provide correct results would have been a better us of viewer's time. But videos like this just state a fact and then handwave around the fact without hitting the core idea. The viewer leaves with a false sense of understanding, and keep wondering about the "bizzare" fact, when it is nothing but good ol' introductory probability. YouTube math needs reform.
  - ndsipa_pomu 2y ago
    I think there's room for Matt Parker's more "fluffy" type of maths presentations as well as the more in depth Grant Sanderson's visualisations. Matt is on a mission to popularise maths and it's easy for anyone that gets inspired/intrigued by a topic to dig deeper. Rather than a reform, maybe there's room for more maths-based videos to fill the gap between recreational and more serious maths?

dahart 2y ago

Either I haven’t seen this before, or forgot it, but it’s surprising because I use the sum of independent uniform variables every once in a while — the sum of two vars is a tent function, the sum of three is a smooth piecewise quadratic lump, and the sum of many tends toward a normal distribution. And the distribution is easy calculated as the convolution of the input box functions (uniform variables). Looking it up just now I learned the sum of uniform variables is called an Irwin-Hall distribution (aka uniform sum distribution). The min of two random vars has the opposite effect as the max does in this video. And now I’m curious - if we use the function definition of min/max — the nth root of the sum of the nth powers of the arguments — there is a continuum from min to sum to max, right? Are there useful applications of this generalized distribution? Does it already have a name?

max_likelihood 2y ago
Perhaps you are thinking of Order Statistics? https://en.wikipedia.org/wiki/Order_statistic https://en.wikipedia.org/wiki/Order_statistic
- dahart 2y ago
  Ah, fascinating, I’ve never used Order Statistics. It doesn’t look exactly like what I was thinking, but there is also a continuum from min to median to max, similar to min to mean/sum to max. I’m not sure but I might guess that for the special case of a set of independent uniform variables, the median and the mean distributions are the same? Does this mean there’s a strong or conceptual connection between the Bates distribution and the Beta distribution? (Neither Wikipedia page mentions the other.) Maybe Order Statistics are more applicable & useful than what I imagined…
  - falseprofit 2y ago
    Median and mean are not the same distribution. Consider three uniform values. For the median to be small two need to be small, but the mean needs three. I think order statistics are more useful than what you described, because “min” and “max” are themselves quantiles and more conceptually similar to “median” than to “mean”. Trying to imagine how to bridge from min/max to mean, I guess you could take weighted averages with weights determined by order, but I can’t think of a canonical way to do that.

keithalewis 2y ago

Front page material? P(max{X_1, X_2} <= x) = P(X_1 <= x, X_2 <= x) = P(X_1 <= x) P(X_2 <= x) = xx. P(sqrt(X_3) <= x) = P(X_3 <= x^2) = x^2. It is late in the day when midgets cast long shadows.

refulgentis 2y ago
I felt the same way, came here to decide whether to comment something negative, saw your comment was just posted. But I first read the top comment from 2 hours ago, apparently this is news-ish to stats people because it's counterintuitive to common methods? shrug

prof-dr-ir 2y ago

If X1...Xn are independently uniformly distributed between 0 and 1 then: P(max(X1 ... Xn) < x) = P(X1 < x and X2 < x ... and Xn < x) = P(X1 < x) P(X2 < x) ... P(Xn < x) = x^n Also, P(X^{1/n} < x) = P(X < x^n) = x^n I guess I am just an old man yelling at clouds, but it seems so strange to me that one would bother checking this with a numerical simulation. Is this a common way to think about, or teach, mathematics to computer scientists?

coliveira 2y ago
> it seems so strange to me that one would bother checking this with a numerical simulation I believe that some people know programming but have little experience with mathematics, so the first thing they'll think about is to "check" numerically that something is true. Which in reality doesn't prove anything, so people should better spend the time to learn some math for these situations.
- ValentinA23 2y ago
  "you can't learn maths on your own, you need a master" My math teacher during my second year in university, who also happened to be a chaos theorist working on cool stuff such as cryptography via chaos synchronization. He was by far the worst teacher I ever had in terms of mental calculation abilities, but he was also the more advanced. I remember a conversation where he explained how he would always implement his algorithms at least twice, on entirely different software and hardware stacks.
- mistercow 2y ago
  I have, on many occasions, thought that I had an analytical solution to a problem pinned down, then checked it numerically and found that I'd made a mistake. It seems weird that there's any hostility towards a perfectly useful tool for checking your work.
  - monadINtop 2y ago
    Not the person you're replying to but I would assume their point was to highlight the distinguishing feature of mathematics - the fact that it's not only possible to completely prove something to be true, but in fact is the only thing you can actually do (yes I know about experimental math etc). But yeah that doesn't necessarily mean anything for pedagogy or just having fun and so on.

gxs 2y ago

Just a side comment on what a great little video. Short, to the point, and the illustrations/animations actually helped convey the message. Would be super cool if someone could recommend some social media account/channel with collections of similar quality videos (for any field).

ajot 2y ago
Well, that YouTube channel, 3Blue1Brown, is amazing. You'll see lots of great explanations with this kind of animations there.