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Suppose we have random variables $X_1, \dots, X_N$, with joint probability distribution $F_{X_1,\dots,X_N}$. Under what conditions does the following equality holds?

$$E[f(X_i)]=E[f(X_j)],\ \ i\neq j.$$

If $X_i$'s are completely dependent, then it trivially holds. Can we generalize it to just saying they are identical (irrespective of whether or not they are independent)?

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    $\begingroup$ This holds for all (measurable) $f$ if $X_1,\dots,X_N$ has the same marginal distributions, such as when $X_1,\dots,X_N$ are exchangeable. $\endgroup$ – user10354138 Apr 18 at 14:42

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