Of course, for completeness, should you want an exact expression and sample directly from the resulting distribution, that is also possible, albeit with substantially more effort. Typically the expression for the quotient of two random variates (and/or its asymptotic expansion) is obtained through inverse Mellin transforms (see https://projecteuclid.org/download/pdf_1/euclid.aoms/1177730201 for a brief overview). You would then implement a custom distribution family for X/Y, and sample directly from that, instead of sampling from X and Y separately.
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