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

18 Actuarial Modelling of Claim Counts
claims in a sufficiently small time interval is negligible when compared to the probability
that he reports zero or only one claim.
Link with the Poisson Distribution
The Poisson process is intimately linked to the Poisson distribution, as precisely stated in
the next result.
Property 1.1 For any Poisson process, the number of events in any interval of length t is
Poisson distributed with mean t, that is, for all s t ≥ 0,
PrNt + s − Ns = n = exp−t tn n= 0 1 2
n!
Proof Without loss of generality, we only have to prove that Nt ∼ oit. For any
integer k, let us denote p k t = PrNt = k, t ≥ 0. The announced result for k = 0 comes
from
p 0 t + t = PrNt = 0 and Nt + t − Nt = 0
= PrNt = 0 PrNt + t − Nt = 0
= p 0 tp 0 t
= p 0 t ( 1 − t + ot )
where the joint probability factors into two terms since the increments of a Poisson process
are independent random variables. This gives
p 0 t + t − p 0 t
t
Taking the limit for t ↘ 0 yields
=−p 0 t + ot p
t 0 t
d
dt p 0t =−p 0 t
This differential equation with the initial condition p 0 0 = 1 admits the solution
p 0 t = exp−t (1.18)
which is in fact the oit probability mass function evaluated at the origin.
For k ≥ 1, let us write
p k t + t = PrNt + t = k
= PrNt + t = kNt = k PrNt = k
+ PrNt + t = kNt = k − 1 PrNt = k − 1
k∑
+ PrNt + t = kNt = k − j PrNt = k − j
j=2