Suppose a sample of size one is drawn from the distribution :

( ) 1( ) 1 ( ) ( 1)

2

f x | | x| exp | x | x ,

= , < < ,

where ( ) is the gamma function.

a. Find the most powerful test for the null hypothesis =1 against the

alternative hypothesis = 2 with size 0 < <1.

b. Is the test in (a) uniformly most powerful ? Explain your answer carefully.

c. Express the power of the test as a function of .