$$\int_{[0,1]^n} \max(x_1,\ldots,x_n) \, dx_1\cdots dx_n$$
My work:
I know that because all $x_k$ are symmetrical I can assume that $1\geq x_1 \geq \cdots \geq x_n\geq 0$ and multiply the answer by $n!$ so we get that $\max(x_1\ldots,x_n)=x_1$ and the integral that we want to calculate is
$n!\int_0^1 x_1 \, dx_1 \int_0^{x_1}dx_2\cdots\int_0^{x_{n-1}} \, dx_n$ and now it should be easier but I'm stuck..
Can anyone help?
Best Answer
This is an alternative approach.
Let $X_i$ ($i=1,\cdots , n$) be independent uniform random variable in $[0,1]$.
What is the PDF of $M=\max (X_1, \cdots, X_n)$?
Then what is $\mathbf{E}[M]$?