Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I would like to see an algebraic proof that the probability mass function of the hypergeometric distribution sums to 1. I know how to prove it combinatorially.

My attempt:

Given: $P(X=x) = \frac{\binom{M}{x} \binom{N-M}{K-x}}{N \choose K}, 0 \leq x \leq K$

I basically need to prove what's below:

$N \choose K$ = $M \choose 0$$N-M\choose K$ + $M \choose 1$$N-M\choose K-1$ $ + \ldots +$ $M \choose K$$N-M\choose 0$

The only problem is that I assume the conclusion when proving the hypothesis! This is flawed. For instance, I am assuming that the sum equals 1 in order to prove that it equals 1.

Can you guys help me?

share|improve this question

1 Answer 1

up vote 2 down vote accepted

Often, you want to use

$$\binom{x}{y} = \binom{x-1}{y} + \binom{x-1}{y+1}$$

for these sorts of things. Eyeballing it, it looks like you just start with $\binom{N}{K}$ and repeat this until you have your equation.


Alternatively, when you know the equation to prove, induction often works quite easily for these. The hard part is usually finding the equation.

share|improve this answer
    
Or, use Vandermonde's identity. –  Sush May 29 at 8:15

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.