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

Mixed Poisson Models for Claim Numbers 33
Computation of the Probability Mass Function
The probability mass at the origin is
( 1
(
N 0 = PrN = 0 = exp 1 − √ 1 + 2) )
Now, taking the derivatives of N with respect to t, and evaluating it at 0 gives the probability
mass function for positive integers. Specifically,
′ N
0 = PrN = 1
∣
= √ N z
1 − 2z − 1
=
√
1 + 2
PrN = 0
∣
z=0
and
′ N
0 = 2PrN = 2
2
∣
= ( ) 3/2
N z∣ + √ ′ N z ∣∣z=0
1 − 2z − 1 z=0 1 − 2z − 1
=
=
2
( ) 3/2
PrN = 0 + √ PrN = 1
1 + 2 1 + 2
√ ( )
2 1 + 2
2
( ) 3/2
PrN = 1 + √ PrN = 0
1 + 2
1 + 2
=
2
PrN = 1 + PrN = 0
1 + 2 1 + 2
In general, we have the following recursive formula
PrN = n =
2
1 + 2
+
(
1 − 3 )
PrN = n − 1
2n
2
PrN = n − 2 (1.43)
1 + 2nn − 1
valid for n = 2 3 4, which allows us to compute the probability mass function of the
Poisson-Inverse Gaussian distribution. The formal proof of (1.43) is based on properties of
the modified Bessel function.
1.4.7 Poisson-LogNormal Distribution
In addition to the Gamma and Inverse Gaussian distributions to model , the LogNormal
distribution is often used in biostatistical studies.