Let Xn be a sample mean in a random sample X1, ...,Xn from POI() distribution and let Tn = n i=1 Xi. Use the Delta

Let Xn be a sample mean in a random sample X1, ...,Xn from POI(λ) distribution and let Tn = n i=1 Xi. Use the Delta method to find the limiting distribution of:

(i) g(Xn) = √n(Xn − λ). (ii) g(Xn)=1/Xn. (iii) g(Tn) = log(Tn). (iv) g(Tn 

) =

Tn. [Hint: Tn/n has approximately N(λ,λ/n) distribution (why?).]