A company insures 3 cities J K and L Losses occurring in the

A company insures 3 cities: J, K, and L. Losses occurring in these cities are independent. The moment generating function for each is as follows:

J = (1-2t)^-3

K = (1-2t)^-2.5

L = (1-2t)^-4.5

X represent combined losses from 3 cities. Calculate (X^3)

Now my first thought was to take 3rd derivative of each separately and evaluate at 0. Then add together. This gives 2,082. Which is incorrect.

The solution was to first multiply the mgf\'s of J, K, and L to get (1-2t)^-10 and then evaluate third deriative at 0 to get 10,560.

My question is why do we multiply the mgf\'s together first? Why was solving each separately and then adding incorrect?

Solution

Because the MGF of a sum of independents = product of their MGFs
(M[X=J + K + L]) = M(J)*M(K)*M(L)
in this case you\'ve already been given them.

This means you have to(in this case) algebraically add the exponents(bases are the same). this yields:

(1-2t)^-10

remember that Mx(T) = E(ext); if T=J + K + L then we have : E(ex(J+K+L)) = E(exJ) * E(exK) * E(exL);