# When does $E[f(X_i)]=E[f(X_j)], i\neq j$?

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)?

• 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. – user10354138 Apr 18 at 14:42