# expectation example

The contraharmonic mean of several positive numbers $u_{1}$, $u_{2}$, $\ldots$, $u_{n}$ is defined as

 $c\;:=\;\frac{u_{1}^{2}\!+\!u_{2}^{2}\!+\ldots+\!u_{n}^{2}}{u_{1}\!+\!u_{2}\!+% \ldots+\!u_{n}}.$

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

If $\langle u_{1}$, $u_{2}$, $\ldots$, $u_{n}\rangle$ is the distribution of the seats of $n$ parties, $s$ the total number of seats in the body of delegates ($u_{1}\!+\!u_{2}\!+\ldots+\!u_{n}\,=\,s$), and one draws a random seat (with probability $1/s$), then the size of the drawn delegate’s party has the expected value

 $\frac{u_{1}}{s}\!\cdot\!u_{1}+\frac{u_{2}}{s}\!\cdot\!u_{2}+\ldots+\frac{u_{n}% }{s}\!\cdot\!u_{n}\;=\;c.$

## References

• 1 Caulier, Jean-François: The interpretation of the Laakso–Taagepera effective number of parties.  – Documents de travail du Centre d’Economie de la Sorbonne (2011.06).
Title expectation example ExpectationExample 2013-11-05 17:59:13 2013-11-05 17:59:13 pahio (2872) pahio (2872) 2 pahio (2872) Example msc 05A18 msc 26E60 msc 60A05