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2. Any density p0 generates a location family {p15 : ,u, E R}, where p,\" is the density of u + Z where Z w pg. There's a simple representation for a draw from any mixture taken over a location family: with \"mixing density\" 71' over the parameter a, the mixture distribution is equal to the distribution of X + Z where X N 1r and Z N pg are independent. (a) Let (1)0 denote the Normal location family {N (a, 02) : ,u E R} (with a xed (I). Consider taking a mixture over this family using density on a that is also Normal. We'll denote this mixing density by 7T, and denote its expectation and variance by 6 and 72. Find resulting the mixture distribution. [Hint: In other words, nd the distribution of X + Z , where X N N (6,72) and Z N N (0, 02) are independent] (b) With prior, model, and data as in Homework 8 Problem 3, nd the [posterior] distribution of the next observation X10