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This might be a naive question, but could someone please tell me how the change-of-variables technique applies to conditional cases? My intuition tells me that there is no difference.

The change-of-variables technique is usually formulated for non-conditional cases, like for finding the joint distribution of $(g_1(X_1,X_2),g_2(X_1,X_2))$ where the joint distribution of $(X_1,X_2)$ is known. My question is: is the technique the same if the conditional distribution $(X_1,X_2|Y=y)$ is known and we want to know the conditional distribution $(g_1(X_1,X_2),g_2(X_1,X_2)|Y=y)$?

$Y$ is a random variable that is not independent from $(X_1,X_2)$, it could be a function of $(X_1,X_2)$. My intuition tells me that $y$ would be treated as a constant and the technique would be the same.

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