please provide detail solution, thanks

2. Any density p0 generates a location family {pM : a 6 R}, where p\" is the density of ,u + Z where Z ~ p0. There's a simple representation for a draw from any mimture taken over a location family: with \"mixing density\" 7r over the parameter a, the mixture distribution is equal to the distribution of X + Z where X N 77 and Z ~ 390 are independent. (a) Let (1)0 denote the Normal location family {N (,u, 02) : a e R} (with a xed 0). Consider taking a mixture over this family using density on ,u that is also Normal. We'll denote this mixing density by 1r, and denote its expectation and variance by 6 and 72. Find resulting the mixture distribution. [Hintz In other words, nd the distribution of X + Z , where X N N (6, 72) and Z ~ N (0,02) are independent]