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Given that I can simulate $X \sim Beta(a,b)$ on $(0,1)$, how can I simulate $Y \sim GBeta(a,b)$ (generalized beta) on $(p, q)$ for arbitrary $p, q \in \mathbb{R}$? Is it just $Y = (p-q) X + p$

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This is non-standard beta distribution. Simply take

$$ Y = X \times (q - p) + p $$

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