Can you provide an example of a joint distribution of $X,Y$ exchangeable random variables for which $\operatorname{E}[X/Y]<\infty$ with $X,Y\sim U(0,1)$ and $P(X=Y)=0$.
I've tried to find one but i think that $\operatorname{E}[X/Y]$ exists only if $P(X=Y)=1$ otherwise the integral diverges and the expectation does not exist.
\begin{align} X \sim {} &\operatorname{Uniform}(0,1) \\[10pt] Y = {} & \begin{cases} \frac 3 2 - X & \text{if } \frac 1 2 < X< 1 \\[10pt] \frac 3 4 - X & \text{if } \frac 1 4 < X < \frac 1 2 \\[10pt] \frac 3 8 - X & \text{if } \frac 1 8 < X < \frac 1 4 \\[10pt] \text{and so on.} \end{cases} \end{align} Then $Y$ also has the same uniform distribution as $X,$ and the distribution of $(Y,X)$ is the same as that of $(X,Y)$ (exchangeability) and the values of $X/Y$ and $Y/X$ remain between $1/2$ and $2$ and so have finite expectation, and $\Pr(X=Y)=0.$