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Stat 5132 (Geyer) Old Second Midterm Solutions

# Stat 5132 (Geyer) Old Second Midterm Solutions - Up Stat...

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Up: Stat 5132 Midterm 2 Problem 1 (a) This is a special case of homework problem 8-4 in the notes. The problem statement says that where . (Here and s are fixed numbers calculated from the data, and denotes a random variable with the posterior distribution of given the data.) The distribution of T is symmetric about zero, so E ( T ) = 0 when the expectation exists (which the problem says to assume). Thus by linearity of expectation . (b) Same as part (a). When a distribution has a center of symmetry, then the center of symmetry is the the median as well the mean when the mean exists. Thus is also the posterior median. Problem 2 (a) The likelihood is The prior is omitting constants that are functions of and but not (normalizing constants of the prior cancel out in calculating the posterior). Thus the unnormalized posterior is

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This is an unnormalized density. Thus that is the posterior distribution of given X . (b) The mean of a distribution is a / b (equation (7) on p. 174 in Lindgren). Hence the mean here is Problem 3 (a) Part (a) of this problem is essentially the same as Problem 3 on the first midterm. The p. d. f. in this problem
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Stat 5132 (Geyer) Old Second Midterm Solutions - Up Stat...

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