# Lower bound for tail of hypergeometric distribution

There are several simple and widely used upper bounds on the tail of the hypergeometric distribution, including $P(X > E[X]+tn) <= e^{-2t^{2}n}$, where X is hypergeometric with parameters N, M, and n. (Thinking of the hypergeometric as describing sampling from a population, N is the population size, M is the number of "interesting" items, n is the size of the sample we draw, and X is the number of interesting items in the sample.) This amusing paper is a good summary:

Matthew Skala. Hypergeometric tail inequalities: ending the insanity, 2009.

It appears to be an unpublished manuscript, but is available at http://ansuz.sooke.bc.ca/professional/hypergeometric.pdf

However, I've been unable to find a simple and reasonably tight lower bound on that same tail probability. Anyone know one?

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I would have thought you can't do much better than taking the maximum of the sum. i.e. $P(X>E[X]+tn)\geq P(X=k)$ where $k$ is the smallest integer above $E[X]+tn$ (for $t>0$) –  probabilityislogic Aug 23 '11 at 15:55
You may be right, but the tail of a hypergeometric (especially for big N) behaves so nicely it seems like there ought to be a simple and fairly tight lower bound. –  DavidDLewis Aug 24 '11 at 12:37