Could be, but it is also possible I'm missing something somewhere, especially because I can't explain why min(random(), random()) looks different from random(random()) or random()*random().

According to the internet what you wrote is the distribution of min(random(), random()), but the distribution of the product is different.

Product: https://math.stackexchange.com/questions/659254/product-dist...

Minimum: https://stats.stackexchange.com/questions/220/how-is-the-min...

I can't explain it either, but I also don't think there is a reason they should look the same.

An attempt to explain the difference without just calculating the things:

random(random()) has an identical distribution to random()*random() (it may even behave identically for a given rng state), although this is different to Math.pow(random(),2) since in that case there's 100% correlation between both parts which makes the expected value product bigger.

random(random()) is also distributed equivalently to

    x=random()
    y=random()
    while(y>=x) y=random()
    return min(x,y)

(the last line could also read 'return y') Comparing that to min(random(),random()), we can see that if the second call to random is smaller than the first, they will return the same result; otherwise, the program equivalent to random(random()) will return a smaller value, therefore the expected value of random(random()) must be lower that that of min(random(),random()).

Ah I think I see it now, yes given X and Y with uniform distribution then the distribution of X is uniform given Y and X < Y, but the distribution of min(X,Y) is not uniform given max(X,Y).

Nice argument! And to clarify further, the equivalent implementation for `random(random())` could be viewed as a rejection-sampling implementation of random(random()).

Another intuitive rephrasing is that while both min(random(), random()) and random(random()) are guaranteed to be at most your first draw, with random(random()) your second draw is constrained to be within the interval of your first, whereas with min(random(), random()) there's no such constraint on the second draw. Thinking about P(x > y); x,y~U should then provide the illumination.

1-random(random()) aka 1-random()*random(),looks slightly skewed toward circumference, and 1-random()^2 looks slightly skewed toward center.

If you plot their difference, 1-random(random()) aka 1-random()*random() is empirically skewed slightly larger than 1-random()^2, but I don't know how to analyze it exactly analytically.