What Is the Distribution of \( n \times \min(U_1, \dots, U_n) \)?

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Homework Statement


Let {U_k}_{k \in \mathbb{N}} be i.i.d from the uniform (0,1) distribution.
I need a formula for the cumulative distribution function of X_n, defined as
X_n := n* \min(U_1, \ldots ,U_n)

Also some advice for X_n := \sqrt{n}* \min(U_1, \ldots ,U_n) would be appreciated.

* is meant to be multiplication..

Homework Equations


The Attempt at a Solution


Know already that P(min(U_1, \ldots ,U_n) \leq x) = 1 - (1-x)^n
 
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Sorry to bump this thread,but I still haven't figured it out. Does anyone know how to find it?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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