If I have a sample $(X_1,Y_1),\dots,(X_n,Y_n)$, is it possible for the sample to be drawn independently, and yet have some of the $X$'s be correlated? If so, I'm trying to understand how a sample can be independent and yet it is possible to have a situation where random variable $X_i$ and $X_j$ be not independent.
I was reading this, where the distribution of $X$ is stated as being arbitrary, and the middle of page 2 states that the "samples are all independent". Here no explicit statement was made regarding the correlation between random variables $X_i$ and $X_j$ in different observations in a sample, so I'm unsure if the statement that "samples are all independent" implies that $X_i$ and $X_j$ are independent. In a later lecture in this series (page 1 here), this sample was used in defining the likelihood as $\Pi p(y_i|x_i;b_0,b_1,s^2)$.
From a different source here, at the bottom of page 4 (in the Conditional Likelihood section), the notes state that for data consisting of $(x_i,y_i)$ pairs, "we do not need to assume that the $x_i$ are independent in order to justify the conditional likelihood being a product; we just need to assume that the $y_i$ are independent when each is conditioned on its own $x_i$". The likelihood function defined here has the same form as my previous source.
I'm not sure how the statement in my second source fits in with the first. I'm also not sure if the correlation between $X_i$ and $X_j$ matters (at least in linear regression).