Let's think about a question:
Suppose $a_1 ,a_2, \cdots,a_n\in\mathbb{R^+}$, such that $a_1 +a_2+ \cdots+a_n=1$. Then randomly select a set of non-negative real numbers:$b_1 ,b_2, \cdots,b_n$, such that $b_1 +b_2+ \cdots+b_n=1$, then what's the expectation of $\sum_{i,j=1}^{n}\mathbf{1}_{a_i\leq b_j}$?
Where $\mathbf{1}_{a_i\leq b_j}$ stands for the flag function: $\mathbf{1}_{a_i\leq b_j}=\begin{cases}1,a_i\leq b_j\\0,a_i<b_j\end{cases}$.
Before calculating, I wonder that: What is "random selection", is it randomly select a point on the hyperplane:$x_1+x_2+\cdots+x_n=1$, or randomly select a vector $\boldsymbol{x}=(x_1,x_2,\cdots,x_n)\in \{(x_1,x_2,\cdots,x_n)|x_i\geq0,|\boldsymbol{x}|=1 \}$, then form the set $(\frac{x_1}{X},\frac{x_2}{X},\cdots,\frac{x_n}{X})$, in which $X=\sum_{i=1}^{n}x_i$.
The latter plan also means "randomly" select a point on unit sphere. So which one is the real "random"? Due to the geometric definition, again, what's "randomly select" a point on the hyperplane or unit sphere?