Symmetric distributions.

(i) LetP be any family of distributions of X = (X1,..., Xn) which are symmetric in the sense that P Xi1 ,..., Xin



∈ A

= P {(X1,..., Xn) ∈ A}

for all Borel sets A and all permutations (i1,...,in) of (1,..., n). Then the statistic T of Example 2.4.1 is sufficient for P, and the formula given in the first part of the example for the conditional expectation E[ f (X) | T (x)] is valid.

(ii) The statistic Y of