questions 8 7- Let X,, X, ,...,X, be Poisson random variables with parameter 2. Assume that 2 has a Gamma(a, ß) prior.(a) Compute the posterior distribution of 2.(b) Obtain the Bayes estimate of A.(c) Compare the MLE of 1 with the Bayes estimate of l.(d) Which of the two estimates is better? Why?8- Let X1 .,X, be a random sample from a Poisson distribution with parameter 2. Show that thesample mean X is sufficient for 2.9- Let X, ,...,X„ be a random sample from a population with pdf0 < x < 1, a > 0f(x) = f(x) =0,%3DotherwiseIs the method of moments estimator for a consistent?10- Let X,,..., X,,n > 4, be a random sample from a population with a mean µ and variance o².Consider the following three estimators of µ:ô =7(X, + 2X2 + 5X3 + X4)Ôz = X1 +X2 +-(X3 + ……·+ Xn-1) + n- 3)5(nÔz = X(a) Show that each of the three estimators is unbiased.(b) Find e(@2, ô1), e(@3, ô1) and e(Ô3, Ô2).11- Let X, ,...,X, be a random sample from the Weibull density(2x-x²/ax > 0f(x) = f(x) =0,otherwiseFind an UMVUE for a.

Name: questions 8 7- Let X,, X, ,...,X, be Poisson random variables with par
Uploaded: 2024-03-05T11:53:49.000Z
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Description: questions 8 7- Let X,, X, ,...,X, be Poisson random variables with parameter 2. Assume that 2 has a Gamma(a, ß) prior.(a) Compute the posterior distribution of 2.(b) Obtain the Bayes estimate of A.(c) Compare the MLE of 1 with the Bayes estimate of l