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The classic birthday paradox considers all $n$ possible choices to be equally likely (i.e. every day is chosen with probability $1/n$) and once $\Omega(\sqrt{n})$ days are chosen, the probability of $2$ being the same, is a constant. I'm wondering if someone could point me to an analysis that also works for a non-uniform distribution of days?

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    In Exercise 13.7 o$f$ *The Cauchy-Schwarz Master Class*, J. Michael Steele uses Schur convexity to show that uniform probabilities are least likely to give birthday matches. So non-uniform birthdays give us a better chance of an early match.2012-08-01

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Maybe those can help you (yes, I know this thread is old, but maybe the answer can be useful to someone else)

http://eprint.iacr.org/2010/616.pdf

http://www.ism.ac.jp/editsec/aism/pdf/044_3_0479.pdf