proof of expected value of the hypergeometric distribution

We will first prove a useful property of binomial coefficientsMathworldPlanetmath. We know


This can be transformed to

(nk)=nk(n-1)!(k-1)!(n-1-(k-1))!=nk(n-1k-1). (1)

Now we can start with the definition of the expected valueMathworldPlanetmath:


Since for x=0 we add a 0 in this we can say


Applying equation (1) we get:


Setting l:=x-1 we get:


The sum in this equation is 1 as it is the sum over all probabilities of a hypergeometric distributionMathworldPlanetmath. Therefore we have

Title proof of expected value of the hypergeometric distribution
Canonical name ProofOfExpectedValueOfTheHypergeometricDistribution
Date of creation 2013-03-22 13:27:44
Last modified on 2013-03-22 13:27:44
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 8
Author mathwizard (128)
Entry type Proof
Classification msc 62E15