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Verification of poisson approximation to hypergeometric distribution


Oct 7, 2013
How can I verify that

$\lim_{N,M,K \to \infty, \frac{M}{N} \to 0, \frac{KM}{N} \to \lambda} \frac{\binom{M}{x}\binom{N-M}{K-x}}{\binom{N}{K}} = \frac{\lambda^x}{x!}e^{-\lambda}$,

**without** using **Stirling's formula** or the **Poisson approximation to the Binomial**?

I have been stuck on this problem for a while, because I don't know how to divide up the terms and factorials without using the help of prior results!

Any help would be appreciated. Thanks in advance.