The equality in distribution X d = Y of random variables is equivalent to E[f(X)] = E[f(Y )] for any nonnegative function f on the set of values of the variables. This equivalence also holds for...

The equality in distribution X d = Y of random variables is equivalent to E[f(X)] = E[f(Y )] for any nonnegative function f on the set of values of the variables. This equivalence also holds for stochastic processes X and Y . Analogous equivalences hold for conditional expectations. In particular, for discrete random variables Y , Y ∗, Z and Z∗ in S,show that the following statements are equivalent: