Let X1,...,Xn be a sample from a Poisson distribution.

i) Find the likelihood ratio test statistic for

H0:=0 vs. HA:=1

where 1>0

ii) Use the fact that the sum of independent Poisson random variables is also Poisson

to explain how to find a rejection region for a test at level

iii) Show that this test is UMP (uniformly most powerful) for testing

H0:=0 vs. HA:>0