A faculty member of a university receives a number of emails. If X represents the number of spam emails in n emails and follows a binomial distribution with parameters  where  is the probability of getting spam email, then find the posterior distribution of considering the following beta distribution.

Also, find the posterior mean of θ .

The following data give the sales of 6 models of mobiles at four different stores. The sales of each mobile (in number of mobiles sold) from each store are given as follows:

A food processing company packages 10 gm of honey in small jars. Previous experience suggests that the volume of a randomly selected jar of the company’s honey is normally distributed with a known variance of 2 gm. Drive likelihood ratio test for testing

at a α level of significance. Given

State whether the following statements are True or False. Give reasons in support of your answer:

(i) If the form of the population is not known and data are in ordinal form then we apply the Wilcoxon Signed rank test for testing hypothesis about average.

(ii) The Neyman-Pearson lemma provides the most powerful test of size a for testing a simple hypothesis against a simple alternative hypothesis.

(iii) The Rao-Blackwell theorem enables us to obtain a manimum variance unbiased estimator through complete statistic.

(iv) For testing goodness of fit when the data are in categorical form, we use K-S test.

(v) In the Bayesian approach, we treat the parameter as a constant.

(b) Differentiate between Rao-Blackwell and Lehmann-Scheffe theorems.

If the number of weekly accidents occurring on a mile stretch of a particular road follows a Poisson distribution with parameter λ . Then

(i) Find the Cramer-Rao lower bound for the variance.

(ii) Also, find the UMVUE of λ .