Suppose the Xi > 0 are independent, and their common density is /( + x)2 for i

Suppose the Xi > 0 are independent, and their common density is

θ/(θ + x)2 for i = 1,...,n, as in example 4. Show that θL

n(θ ) =

−n + 2 n

i=1 Xi/(θ + Xi). Deduce that θ → θL

n(θ ) decreases from n to −n as θ increases from 0 to ∞. Conclude that Ln has a unique maximum. (Reminder: L

n means the derivative not the transpose.)