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 the sample mean X is sufficient for 2. 9- Let X, ,...,X„ be a random sample from a population with pdf 0 < x < 1, a > 0 f(x) = f(x) = 0, %3D otherwise Is 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(n Ôz = 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/a x > 0 f(x) = f(x) = 0, otherwise Find an UMVUE for a.

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Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.