# Solved – Which distribution has its maximum uniformly distributed

Let's consider $$Y_n$$ the max of $$n$$ iid samples $$X_i$$ of the same distribution:

$$Y_n = max(X_1, X_2, …, X_n)$$

Do we know some common distributions for $$X$$ such that $$Y$$ is uniformly distributed $$U(a,b)$$?

I guess we can always "construct a distribution" $$X$$ to enforce this condition for $$Y$$ but I was just wondering if a famous distribution satisfies this condition.

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Let $$F$$ be the CDF of $$X_i$$. We know that the CDF of $$Y$$ is $$G(y) = P(Yleq y)= P(textrm{all } X_ileq y)= prod_i P(X_ileq y) = F(y)^n$$

Now, it's no loss of generality to take $$a=0$$, $$b=1$$, since we can just shift and scale the distribution of $$X$$ to $$[0,,1]$$ and then unshift and unscale the distribution of $$Y$$.

So what does $$F$$ have to be to get $$G(y) =y$$? We need $$F(x)= x^{1/n}I_{[0,1]}$$, so $$f(x)=frac{1}{n}x^{1/n-1}I_{[0,1]}$$, which is a Beta(1/n,1) density.

Let's check

``> r<-replicate(100000, max(rbeta(4,1/4,1))) > hist(r) `` Rate this post