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Given a joint probability density function of two random variables $X$ and $Y$, how can I calculate the conditional probability density function $ f_{x|3y<x} $?

Thanks!

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The formula is: $$f_{[X\,\mid\, 3Y<X]}(x) ~=~ \dfrac{\int\limits_{-\infty}^{x/3}f_{X,Y}(x,t)\;\mathrm d t}{\int\limits_{-\infty}^\infty\int\limits_{-\infty}^{s/3}f_{X,Y}(s,t)\;\mathrm d t\;\mathrm d s}$$


It is analogous to the discrete case.

$$\mathsf P(U=u\mid 3V<U) ~{~=~ \dfrac{\mathsf P(U=u, 3V<u)}{\mathsf P(3V<U)} \\[4ex] ~= ~ \dfrac{\sum\limits_{t<u/3}\mathsf P(U=u, V=t)}{\sum\limits_s\sum\limits_{t<s/3}\mathsf P(U=s, V=t)} }$$

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