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Stat 5132
Midterm 2
Problem 1
(a)
This is a special case of homework problem 84 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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 Spring '02
 Staff
 Normal Distribution, Probability theory, probability density function, Estimation theory, Prediction interval

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