Square roots and maxima
leancrew.com
leancrew.com
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?
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.
Yeah I'm real fun at parties.
The other neat thing Heavy Gear did was they had none of this ablative armor bullshit like you see in Battletech. The armor ether works and you get no damage or it gets penetrated and you get full damage.
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?
P(min{X1, X2, ..., Xn} < x) =
P(X1<x or X2<x ... or Xn < x) =
P(not(not(X1 < x) and not(X2 < x) ... and not(Xn < x))) =
1-P(not(X1 < x) and not(X2 < x) ... and not(Xn < x)) =
1-P(not(X1 < x))⋅P(not(X2 < x)) ... ⋅P(not(Xn < x)) =
1-(1-P(X1 < x))⋅(1-P(X2 < x)) ... ⋅(1-P(Xn < x)) =
1-(1-x)^n
which curve, in the [0, 1]^2 square, is just x^n rotated around (1/2; 1/2) by 180 degrees.
As a probability nerd myself, I was yelling at my browser for him to take the difference between the true CDF and the empirical CDF. I.e., not just the plot where “analysis” CDF overlaps empirical CDF, but the difference between the two, scaled up by some number…say, the square root of the number of samples? ;-)
Then we would have a realization of a discretized Brownian bridge, a kind of rescaled Brownian motion. And then we could have all kinds of fun looking at where the maximum difference falls (it will usually not be near the endpoints), and the size of the set adjacent to the maximum, and the local behavior of the process around the maximum (it’s expectation should be “cusped”, not smooth, although other processes will be smooth there).
Some of those topics are really rather advanced, and they are all accessible by simulation if you follow your nose.
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.
But yeah that doesn't necessarily mean anything for pedagogy or just having fun and so on.
The proof as a platonic ideal is infallible, but in reality, it gets fed into a fallible meat computer, and in practice, even very smart and careful people do make mistakes, often at the individual level, but sometimes as an entire community.
Two famous examples were apparent proofs of the four color theorem in the late 19th century, each of which were widely accepted for over a decade before being shown incorrect.
We have better tools nowadays, obviously, but these still only increase confidence, which is exactly what running simulations does.
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.
In other cases, a numerical simulation giving a wrong answer can quickly tell you that your apparently valid reasoning contains a mistake. That’s really useful, because subtle reasoning errors are really easy to make, and math is full of fun false proofs.
A wrong simulation is strong evidence that you’ve misunderstood something, and so by necessity, a correct simulation is (weaker) evidence that you’ve understood correctly.
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).
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.
Rather than a reform, maybe there's room for more maths-based videos to fill the gap between recreational and more serious maths?