- Question 11 A Researcher Plans To Use A Random Sample X1 Xn Of Size N From A Population With Probability Density 1 (141.31 KiB) Viewed 21 times
)Question 11 ( A researcher plans to use a random sample X1, ..., Xn of size n from a population with probability density function e-Oxi f(3:0) = { for fi > 0 otherwise, 0 where parameter 2 > 0. = (a) Suppose (in this part only) that the researcher has just one observation (X) and wishes to test the null hypothesis H. : 0 = 1 against the alternative hypothesis H1 : 0 = 01 > 1. If the rejection region is X < c and the required type I error probability is a = = 0.1, determine the critical value c. If 01 = 2, compute the probability of a type II error and explain its meaning. (b) If prior distribut for parameter 0 > 0 is (0) ae-a0,0 > 0 and zero elsewhere, where a > 0, show that the posterior distribution h(0|21, ..., &n) is a gamma distribution with parameters a = n +1 and B = a + nī, where T is the sample mean. = 1 (c) What is the formula for the Bayes estimator E(@|1, ..., &n)? If n = 4, T = 2 and a = 1, what is the Bayes estimate of e?