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

16 Actuarial Modelling of Claim Counts
for = nq to be small. It is the largeness of n and the smallness of q = /n that are
important. However most of the data sets analysed in the literature show a small frequency.
This will be the case with motor data sets in insurance applications.
Moments of the Poisson Distribution
If N ∼ oi, then its expected value is given by
Moreover,
EN =
+∑
k=1
= exp−
k exp− k
k!
+∑ k+1
k=0
k!
= (1.14)
EN 2 =
+∑
k=1
so that the variance of N is equal to
= exp−
k 2 exp− k
k!
+∑
k=0
k + 1 k+1
= + 2
k!
VN = EN 2 − 2 = (1.15)
Considering Expressions (1.14) and (1.15), we see that both the mean and the variance of
the Poisson distribution are equal to , a phenomenon termed as equidispersion.
The skewness of N ∼ oi is
N = √ 1 (1.16)
Clearly, N decreases with . For small values of the distribution is very skewed
(asymmetric) but as increases it becomes less skewed and is nearly symmetric by = 15.
Probability Generating Function and Closure Under Convolution for the
Poisson Distribution
The probability generating function of the Poisson distribution has a very simple form.
Coming back to the Equation (1.5) defining N and replacing the p k s with their expression
(1.13) gives
+∑
N z = exp− zk = exp ( z − 1 ) (1.17)
k=0
k!
This shows that the Poisson distribution is closed under convolution. Having independent
random variables N 1 ∼ oi 1 and N 2 ∼ oi 2 , the probability generating function of
the sum N 1 + N 2 is
N1 +N 2
z = N1
z N2
z = exp ( 1 z − 1 ) exp ( 2 z − 1 ) = exp ( 1 + 2 z − 1 )
so that N 1 + N 2 ∼ oi 1 + 2 .