# Birthday paradox for non-uniform distributions

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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Note that such a solution would implicitly then include the solutions for $m<n$ in the even distribution case. In general, I believe you can prove that it cannot take longer with another distribution - that is, the "even distribution" is the worst case in some sense. If $p$ is a probability on $1,...,n$ and $b(p)$ is the birthday number for $p$ - the number at which the odds of two selections being equal out $b(p)$ selections is greater than $1/2$ - then $b(p)$ is maximized when $p$ is the even distribution. –  Thomas Andrews Aug 1 '12 at 18:05
It's not clear what the text "is a constant" applies to above. –  Thomas Andrews Aug 1 '12 at 18:10
In Exercise 13.7 of 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. –  Byron Schmuland Aug 1 '12 at 19:22