the distribution of the maximum and minimum
Of a whole sample, the single largest and single smallest values are the extremes, and they have a satisfyingly simple distribution. How hot was the hottest day this year? What was the weakest link in a chain? These are questions about a maximum and a minimum, and a tiny independence trick gives their cdf directly.
The key observation: the maximum of n values is at most x exactly when EVERY value is at most x. For independent identically distributed X_1, ..., X_n with common cdf F, that means P(max at most x) = F(x)^n — just multiply, because the events are independent. The minimum uses the mirror idea: the minimum is at least x exactly when every value is at least x, so P(min at least x) = (1 - F(x))^n, giving F_min(x) = 1 - (1 - F(x))^n. Differentiate either to get the density. So the extreme distributions follow from the original F by raising to a power — no integration needed.
The behaviour as the sample grows is striking. For the maximum, F(x)^n pushes probability toward the upper end: take more samples and the maximum drifts higher and concentrates near the top of the support. This is the seed of extreme-value theory: suitably rescaled, sample maxima converge to one of just three limiting shapes (Gumbel, Frechet, Weibull), which is why engineers model 100-year floods and record loads with these laws.
Let X_1, ..., X_n be independent uniforms on (0, 1), where F(x) = x. Then P(max at most x) = x^n, so the density of the maximum is n x^(n-1) — heavily weighted toward 1. For n = 10, the maximum has expected value 10/11, already very close to the top of the range.
P(max at most x) = F(x)^n: the max climbs toward the top as the sample grows.
These power formulas assume the sample is independent and identically distributed. With dependence (or different distributions) you cannot simply multiply — the events are no longer independent.