# Sample from a estimated conditional distribution

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 $\hat{f}_{X \mid Y}(x, y)$?

• I guess that by saying "for each X" you imply that you are talking about discrete distribution...? – Tim Aug 4 '16 at 7:28

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.