I'm having a bit of trouble wrapping my head around this one and would appreciate some guidance.
So we have two continuous (interdependent) random variables $X$ and $Y$, distributed uniformly on [0,$\theta$]. I'm looking for the joint density $f(y|X=x;\theta)$ where $\theta$ is a parameter. We also know that $\theta$ is distributed uniformly over [0,2] but not its value. I know that this is equal to the likelihood function $L(\theta;y|X=x)$ correct? I had no trouble calculating $f(y|x)$, but I'm not quite sure how to compute this function directly. There should be a way to apply some kind of Bayesian inference here, no?
This deals with auctions where $\theta$ is the previous winner's bid (fixed in the next round) if anyone is curious. The intuition is that knowing the previous winning bid $\theta$ causes the bidder in the next round to adjust how much he is willing to bid relative to his 'value' of the object.