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
Canonical name	ExpectationExample
Date of creation	2013-11-05 17:59:13
Last modified on	2013-11-05 17:59:13
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	2
Author	pahio (2872)
Entry type	Example
Classification	msc 05A18
Classification	msc 26E60
Classification	msc 60A05