4 Mathematics 341/686 December 8, 2015 1. Let X1 , X2 , . . . , Xn be a random sample from N (1 , 2 ) and let Y1 , Y2 , . . . , Yn be a random sample from N (2 , 2 ) and 2 is known. (a) Find the likelihood ratio for H0 : 1 = 2 VS H1 : 1 = 2. (b) Find the rejection region for the likelihood ratio test. (c) Find the power function. (d) Find a 1- condence interval for 1 2 . 2. Let X be a single observation from the density f (x; ) = x1 for 0 < x < 1 and > 0. (a) Dene Y = logX, show that Y follows an exponential distribution. (b) Evaluate the condence coecient of the set [1/2y, 1/y]. (c) Find a pivotal quantity and use it to set up a condence interval having the same condence coecient as in Part (a). (d) Compare the two condence intervals. 3. X is a single observation from the density f (x; ) = (1 + )x for 0 x 1 and > 1. (a) Find the simple likelihood ratio for H0 : = 0 versus H1 : = 1. (b) Find the rejection region in terms of a constant c where 0 c 1. (c) Find the power function of the likelihood test in (a) in terms of c. (d) Find the size of of test in (a) in terms of c. (e) Type II error is dened as the probability of not rejecting H0 when H1 is true. Find Type II error, t2 , in terms of c. (f) Let t1 be the type I error, nd the proper c value such that 3t1 + t2 is minimized. 4. Let X1 , , Xn be a random sample of from a Poisson() distribution. Find a UMP level test of H0 : 0 VS H1 : > 0 . 1