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In the classic birthday paradox problem we assume a uniform distribution of birthdays across the year, i.e. $p(x)=\frac{1}{365}, x=1,2,...,365$. What is the coincidence probability of two birthdays if we now assume that we sample from a power law ($ p(x) = x^{-k} $), instead of a uniform distribution?

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I suppose you mean $p(x) = x^{-k}/\zeta(k)$ for positive integers $x$, where $\zeta$ is the Riemann zeta function. If $X$ and $Y$ are independently chosen from this distribution, $$ \mathbb P(X=Y) = \sum_{x=1}^\infty p(x)^2 = \frac{\zeta(2k)}{\zeta(k)^2}$$

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