$x\_1 + x\_2 + \dots + x\_n = 1, 0 \leq x\_i \leq 1$, and $(x\_1, x\_2, \dots, x\_n)$ evenly distributes on its restricted space, obviously which is a polygon on $n - 1$ dimension plane.


Let random variable $Z = \max(x\_1, x\_2, \dots, x\_n)$, what is the probability distribution function $F(m) = P(Z < m)$. 


Obviously, $F\left(\frac{1}{n}\right) = 0, F(1) = 1$.


Dose this problem have a analytical solution? If not, Dose this problem have a recursive solution?



What the probability of the max value of restricted random variable?
It is now a textbook fact that the joint distribution of your random variables $x\_1,\dots,x\_n$ is the same as that of $R\_1,\dots,R\_n$, where $R\_i:=H\_i/(H\_1+\dots+H\_n)$ and the $H\_i$'s are iid standard exponential random variables. [Moran, page 93](https://www.jstor.org/stable/2983572?seq=1#page_scan_tab_contents) ascribes mentioning of this fact to [Fisher](https://www.jstor.org/stable/95247), and a proof of it -- without a specific reference -- to Clifford. 


[Fisher](https://www.jstor.org/stable/95247) also gave the distribution of $\max\_i R\_i$, which is, by what was just stated, the same as the distribution of $\max\_i x\_i$: 
$$P(\max\_i x\_i<u)=P(\max\_i R\_i<u)
=\sum\_{j=0}^n(-1)^j\binom nj(1-ju)\_+^{n-1}
$$
for $u\in[0,1]$, where $v\_+:=\max(0,v)$. 


