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Homework Statement
Let [itex](S, \Sigma, P)[/itex] be a probability space. Let X and Y be two random variables on S that satisfy [itex]P \circ X^{-1} = P \circ Y^{-1}[/itex] (i.e. they are identically distributed) and such that E(X), E(Y), var(X) and var(Y) exist and are finite. Prove that E(X) = E(Y) and var(X) = var(Y).
The attempt at a solution
I can only think of proving this the long and tedious way: first when X and Y are simple, then for X and Y bounded, then for X and Y nonnegative, and then finally for X and Y integrable. Is there an easier way?
Let [itex](S, \Sigma, P)[/itex] be a probability space. Let X and Y be two random variables on S that satisfy [itex]P \circ X^{-1} = P \circ Y^{-1}[/itex] (i.e. they are identically distributed) and such that E(X), E(Y), var(X) and var(Y) exist and are finite. Prove that E(X) = E(Y) and var(X) = var(Y).
The attempt at a solution
I can only think of proving this the long and tedious way: first when X and Y are simple, then for X and Y bounded, then for X and Y nonnegative, and then finally for X and Y integrable. Is there an easier way?