## Transcribed Text

1. Suppose that Y,,Y2, Yn is an iid random sample from an exponential
distribution with probability density function
f(y10)=8e 7, y>0
Suppose that the prior distribution of the parameter 0 is gamma with parameters
a
and That is, the prior density function of 0 is given by
1 ,
0>0
Find the corresponding posterior distribution of the parameter A. Find the
posterior mean and show that it can be written as a weighted average of the prior
mean and an estimate of 0 from only the data.
1
2.
If Y10,3 - Beta(0,3), write down the probability density function and
find a corresponding kernel. The kernel should be as simple as possible.
3. Suppose that the random variable Y has the inverted Pareto distribution with
density function
8>0,0>0
0,
lsewhere
If
8 = 1, show that the probability density function given above can be written as
an exponential family
2

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