Dear All Hope you are well I find this question very confusing at the set up Solution sets out N|Phi=Theta ~Binomial (3,theta) , why is Phi= Theta? are we using this interchangeably? I get N the number of claims are being told to be Binomial But i am confused with this set up : ie ' N with parameter Phi follows a binomial distribution with parameters n = 3 and p = Theta The parameter Phi has the pdf .... which involves Theta. Then why do we set Phi = Theta? A more intuitive set up is from Page 39 of CH18, where we have loss ratio depending on theta, loss ratio Xk itself is gamma dist (Vk, Vk/Theta), Theta itself has its own distribution of uniform (50,500). To me this means distribution of Xk depends on theta, so it makes sense to me when working out Beta, small Phi and Lambda and then Z. Thanks !! Jun
Hi Jun You need to be very careful conceptually because there at least three traps. The use of the dummy variables capital letter phi and theta is different from the notes. The exam question is dealing with claim numbers hence the use of the binomial distribution. If you reread the question you will see that the binomial distribution relates to the number of claims for an individual. Ordinarily you would be estimating the benchmark claims ratio and its moments. In this case you are estimating the benchmark claim numbers per individual and its moments. Your claim numbers per individual is essentially your 'claims ratio'.
Hi, i am struggling to understanding the following transition in the solution to this question, could i have some insight into any missing steps please? E[3 Φ(1- Φ)]= ∫ 3θ(1- θ)π(θ)dθ thanks