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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?

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up vote 2 down vote accepted

Often, you 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.

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Or, use Vandermonde's identity. – Silent May 29 '14 at 8:15

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