expectation example

The contraharmonic mean of several positive numbers$u_1$, $u_2$, $\mathrm{\dots }$, $u_n$ is defined as

$$c:=\frac{u_1^2+u_2^2+\mathrm{\dots }+u_n^2}{u_1+u_2+\mathrm{\dots }+u_n}.$$

This has certain applications; oneof them is by [1] the following.

If $⟨u_1$, $u_2$, $\mathrm{\dots }$, $u_n⟩$ is the distribution of the seats of $n$ parties,$s$ the total number of seats in the body of delegates($u_1+u_2+\mathrm{\dots }+u_n=s$),and one draws a random seat (with probability $1/s$), then the size of the drawndelegate’s party has the expected value

$$\frac{u_1}{s}\cdot u_1+\frac{u_2}{s}\cdot u_2+\mathrm{\dots }+\frac{u_n}{s}\cdot u_n=c.$$