Distribution of a uniform random variable over the sample maximum

Question

Suppose $X_1, \ldots X_n \sim U(0,b)$ (i.i.d.), with $b \in (0, \infty)$.

Let $M = \max \{X_1, \ldots X_n\}$. What's the distribution of $X_1 / M$?

Answer 1

Assume that $b=1$ and let $M_2:=\max\{X_2,\ldots,X_n\}$. For $0< x\le 1$, $$ \mathsf{P}(X_1\le xM)=\mathsf{P}(X_1\le xM,X_1> M_2)+\mathsf{P}(X_1\le xM,X_1\le M_2) $$ The first term on the RHS is $\mathsf{P}(X_1> M_2)1\{x=1\}$. The second term equals $$ \mathsf{P}(X_1\le xM_2)=\int_0^1\mathsf{P}(xM_2\ge z)dz=\int_0^x(1-(z/x)^{n-1})dz=\frac{x(n-1)}{n}. $$ Finally, $$ \mathsf{P}(X_1/M\le x)=\frac{1\{x=1\}}{n}+\frac{x(n-1)}{n}. $$