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I want to pick $x+\sqrt{-1}y$ at random with respect to the hyperbolic measure from the standard domain $\{z:|z|\geq1,|\mathcal R(z)|\leq\frac12\}$ for $SL_2(\mathbb Z)\backslash\mathbb H$. What does the probability distribution function of $\frac1{\sqrt y}$ look like?

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