If I have a sample $(X_i,Y_i)_{i=1}^n$ I can estimate the joint CDF by: $$F_{X,Y}(x,y) = \frac{1}{n}\sum_{i=1}^nI[X_i\leq x,Y_i \leq y]$$ Assume now I observe only $(X_i)_{i=1}^n$, but I know that $Y\sim U[0,1]$. How does one estimate the bivariate CDF in this case? I would be thankfull for any reference or suggestion.

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    $\begingroup$ Thought experiment: If you don't observe $X$ and $Y$ together, how can you expect to learn about their dependence relations? What if I secretly replaced your $Y$ by $1 - Y$? Or I divided $[0,1]$ into $n$ equally spaced nonoverlapping blocks, chose a random permutation, then replaced your $Y$ by looking at its value and remapping it to the corresponding relative position in the image under the permutation? What if...(ok, I'll stop there). :-) Similar questions have come up several times on this site, so a search might be helpful. Additionally, you might like to read about copulas. Cheers. :) $\endgroup$ – cardinal Jan 16 '13 at 13:18
  • $\begingroup$ No, it's not naive! It's a good question and one that comes up relatively often (as I hinted at in the previous comment). It's good to think about these things because it not only sharpens statistical and mathematical intuition but also forces us to think about how we use random variables to model our knowledge of an actual physical process. Not understanding this can have serious consequences (though that article somewhat overstates some aspects for "entertainment" value, in my opinion). $\endgroup$ – cardinal Jan 16 '13 at 15:12

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