Suppose we have independent and identically distributed observations y = (y,..., yn), with yi Exponential (A)....

 

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Suppose we have independent and identically distributed observations y = (y₁,..., yn), with yi Exponential (A). (a) Suppose that we specify ~ Gamma(a, ß) as the prior distribution on the unknown model parameter, A. Derive the posterior mean, E(Xly) and posterior standard deviation, Var(\ly). (5 points) (b) Now, suppose that we are interesting in learning about = 1, i.e., the mean of an Exponential (X) distribution. Show that a conjugate prior distribution for is an Inverse-Gamma (a, 6) distribution. You may use that if X Inverse-Gamma (a, ß), then the associated probability density function is Ba p(x|a, ß) = r(a)(¹/x)ª+¹exp(-8/x), defined over the range/support x > 0, with parameters a > 0 and 3 > 0. (5 points) Suppose we have independent and identically distributed observations y = (y₁,..., yn), with yi Exponential (A). (a) Suppose that we specify ~ Gamma(a, ß) as the prior distribution on the unknown model parameter, A. Derive the posterior mean, E(Xly) and posterior standard deviation, Var(\ly). (5 points) (b) Now, suppose that we are interesting in learning about = 1, i.e., the mean of an Exponential (X) distribution. Show that a conjugate prior distribution for is an Inverse-Gamma (a, 6) distribution. You may use that if X Inverse-Gamma (a, ß), then the associated probability density function is Ba p(x|a, ß) = r(a)(¹/x)ª+¹exp(-8/x), defined over the range/support x > 0, with parameters a > 0 and 3 > 0. (5 points)

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