8. Symmetric distributions. (i) Let 9 be any family of distributions of X = (XI " .. , X,,) which are symmetric in the sense that p{ (.\';" ... ,.\';J E A} = P{ (XI' " '' X,,) E A} for all Borel sets A and all permutations (il" .. , i,,) of (1, . . . , n). Then the statistic T of Example 7 is sufficient for 9, and the formula given in the first part of the example for the conditional expectation E[f( X)IT(x)) is valid. (ii) The statistic Y of Problem 5 is sufficient. (iii) Let XI" ' " X" be identically and independently distributed according to a continuous distribution P E 9, and suppose that the distributions of 9 are symmetric with respect to the origin. Let V; = 1.\';1 and IV; = Jl(i) ' Then (WI' . . . , IV;,) is sufficient for 9 .