If I have a set $Y$ of independent random variables and a set X of random variables which are not necessarily independent, and we know that $P_{Y_i | X_i}(y_i | x_i) = e^{2|\theta x_i - y_i|}$, where $\theta$ is not known, how would one go about finding the joint PDF of the y_i's conditioned on the x_i's?
Honestly, I feel like the question is awkwardly posed. I assume it means the joint PDF of $P_{Y_1 | X_1}(y_1 | x_1), P_{Y_2 | X_2}(y_2 | x_2),$ and so on, but even then I would not see how to go about finding any expression that is of any use. I'm inclined to say it's simply the product of each conditional probability, but seeing as only the $y$'s are independent, I'm not sure if that would be true. Any help is appreciated and I apologize for the ambiguity.