Sta601 HW

1. for X1, X2, ... X15 that is a random sample from 
   exp(lambda) distribution, 

a) find a UMP test for 
   H0: lambda <= 1
   Ha: lambda > 1

with alpha = 0.05. Proof the test you find is UMP. 
(you can use any theorem we proved in class). 

b) find a UMPU test for 
   H0: lambda = 1
   Ha: lambda != 1

with alpha = 0.05. 
Proof that the test you find is NOT UMP test. 

2. Lindgran page 356  problem 9.90 and 9.91 

3. Lindgran page 355  problem 9.86 

4. Lindgran page 349  problem 9.73 but for sample size n= 10. 
Is there a UMP test of H0 theta <= 1 vs. Ha: theta > 1 ?
(of size alpha = 0.05)
(If yes, find it and proof it is. If not, show there do not exist
such test).