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Say, there are $\{X_i\}_{i =1}^n$. Also, there is the estimated conditional distribution of $Y$ given $X$, $\hat{f}_{Y \mid X}(y \mid x)$. If I draw a $Y_i$ from $\hat{f}_{Y \mid X}(y \mid x)$ for each $X_i$, is it reasonable to say that $\{Y_i\}_{i = 1}^n$ is a sample from $\hat{f}_Y(y)$? Is it also reasonable to say that $\{X_i , Y_i\}_{i=1}^n$ is a sample from the estimated joint distribution $\hat{f}_{X \mid Y}(x, y)$?

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  • $\begingroup$ I guess that by saying "for each X" you imply that you are talking about discrete distribution...? $\endgroup$
    – Tim
    Aug 4, 2016 at 7:28

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  1. If the $n$ $X$-values are a random sample from the marginal distribution for $X$, then - because $f_{X,Y}(x,y)=f_{Y|X}(y|x)f_X(x)$ - you'll have a sample from the joint distribution and so the $Y_i$ values will in turn be a sample from their marginal.

  2. If the $n$ $X$-values are just some arbitrary collection of X-values (as your question implies), then you won't have a random sample from the marginal distribution for $X$, so you won't have a sample from the joint distribution and so the $Y_i$ values won't in turn be a sample from their marginal.

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  • $\begingroup$ Glen, could you please elaborate a bit more on point 1) ? How do you get the marginal from a sample of the joint distribution? $\endgroup$ Apr 9, 2018 at 21:38
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    $\begingroup$ Sample (X,Y) many times and then do an ecdf, or a histogram, or a KDE (or whatever you like) of whichever of the variables you want to look at the marginal distribution of. When you have a data set that's supposed to be a random sample and you look at a histogram (say) of one of the variables, that's already what you're doing. If you've looked at data, you must have done this many times already. $\endgroup$
    – Glen_b
    Apr 9, 2018 at 23:16

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