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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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