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Comment by dahart | Hacker News Reader

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?

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Perhaps you are thinking of Order Statistics? https://en.wikipedia.org/wiki/Order_statistic

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…

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.

the canonical mapping is via norms. min is the 0 norm, 1 is the mean and the inf norm is maximum

The reason that they do not look the same is that the order statistics are there presented for an exponential function, which has an unbounded upper range. When you do it on a uniform distribution with n variables, you get an n'th power and n'th root at the extremes, with varying lopsided normal-looking distributions in between.

It's called the generalized mean:

https://en.wikipedia.org/wiki/Generalized_mean

Ah yes, this is the functional definition of the min/mean/max continuum, thanks! So does this have applications in statistics, and is it ever used to produce a distribution of random variables?

I build a whole TTRPG around this fact, so that it's easier to create realistic performance curves for characters as they skill up.

Yeah I'm real fun at parties.

One of the things I liked about the Heavy Gear table top game was that the roll mechanic was to roll N dice and pick the highest. where N was your skill level. Now this did make the game somewhat brutal. but there was a lot less of the absurd high skill wiffing you see as in a D & D type system.

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.

Playing around with a somewhat similar mechanic lead to me proposing https://projecteuler.net/problem=240.

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