Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

272 Actuarial Modelling of Claim Counts
type i i = 1m. Then the random variables N 1 N m are independent and Poisson
distributed with respective parameters q 1 q m .
Proof
Since given N = n, N i ∼ inn q i , we can write
PrN i = k =
∑
PrN i = kN = n PrN = n
n=k
( ∑ n
=
)q k i
n=k
k
1 − q i n−k exp− n
n!
= exp− q i k ∑ 1 − q i n
k!
n=0
n!
= exp−q i q i k
k!
which proves that N i ∼ oiq i .
We now prove the independence property of the N i s. To this end, let us show that the
joint probability mass function factors in the product of marginal probability mass functions:
PrN 1 = n 1 N m = n m
n 1+···+n m
= PrN 1 = n 1 N m = n m N = n 1 +···+n m exp−
n 1 +···+n m !
= n 1 +···+n m !
q n 1
1
n 1 !···n m ! ···qn m
m exp− n 1+···+n m
n 1 +···+n m !
m∏
= exp−q j q j n j
j=1
n j !
m∏
= PrN j = n j
j=1
which completes the proof.
□
Let us now denote as q k1 q k2 q km the probability that the claim is of type 1 2m,
respectively, for a policyholder with = k . The identity q k1 +q k2 +···+q km = 1 obviously
holds true. Now, let N 1 N 2 N m be the number of claims of type 1 2m,
respectively. Considering Property 6.1, given and , the random variables N 1 N 2 N m
are mutually independent, with respective conditional probability mass function
PrN l = j = = k = exp− k q kl − kq kl j
j!
j = 0 1
for l = 1m.