What is the simplest example of a non-exchangeable sequence of random variables? I read the wiki article on exchangeability which contained examples, but not negative examples. Can you please provide the simplest negative example of exchangeable random variables?
Also, I know that:
$P(x_1, x_2) = P(x_1 | x_2) P(x_2) = P(x_2 | x_1) P(x_1)$
I thought this is always true, but now I think it is invalid for cases of non-exchangeable random variables, but I cannot see examples.
 A: A list of variables, each with different marginal distributions, is an easy example of non-exchangeable, even if they're independent. Or a sequence where only one variable has a different distribution from the others.
But even variables that are identically distributed can be non-exchangeable. Example: Let the pair $(X_1,X_2)$ be uniformly distributed over the shaded 'pinwheel' region pictured below.

Then both $X_1$ and $X_2$ have the same (uniform) marginal distribution, but they are not exchangeable. Suppose $X_1$ lives on the horizontal axis, and $X_2$ on the vertical, and the enclosing square runs from $(0,0)$ to $(2,2)$. Then $P(X_1\le X_2\le 1)= 1/4$ while $P(X_2\le X_1\le 1)=0$. For $(X_1,X_2)$ to be exchangeable we'd need the joint distribution to be symmetric: it would need to look the same when reflected across the diagonal line $x_2=x_1$.
For a discrete example of non-exchangeable but identically distributed random variables, let the pair $(X_1, X_2)$ take the three values $(0,1)$, $(1,2)$, $(2,0)$ with equal probability. Then both $X_1$ and $X_2$ are uniformly distributed on the set $\{0,1,2\}$. But $P(X_1<X_2) = 2/3$ while $P(X_2 < X_1) = 1/3$, hence $X_1$ and $X_2$ cannot be swapped. (This  construction can be generalized to create a list $X_1, X_2,\ldots, X_n$ of $n$ random variables  that are identically distributed but are not exchangeable: Place mass $1/n$ on the point $(0, 1, \ldots, n-1)$, "rotate" to create the next point $(1, 2,\ldots, n-1, 0)$, and repeat.)
