47. Let Xl" ' " Xn be a sample from the inverse Gaussian distribution I(p., 1') with density / l' exp( _. _1'(X _ p./), 2'1Tx 3 2xp.2 X> 0, 1', P. > O. Show that there exists a UMP test for testing (i) H: p. .s P.o against p. > p.o when l' is known; (ii) H: l' TO against l' > TO when p. is known. In each case give the form of the rejection region. (iii) The distribution of V = T(X; - p.)2/X;p.2 is xl, and hence that of TL[(X; - p.)2/X;p.2] is x~ [Let Y=min(X; ,p.2/X;), Z=T(Y-p.)2/p.2y. Then Z= V and Z is xl [Shuster (1968)].] Note . The UMP test for (ii) is discussed in Chhikara and Folks (1976)