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

10 Actuarial Modelling of Claim Counts
1.3 Poisson Distribution
1.3.1 Counting Random Variables
A discrete random variable X assumes only a finite (or countable) number of values. The
most important subclass of nonnegative discrete random variables is the integer case, where
each observation (outcome) is an integer (typically, the number of claims reported to the
company). More precisely, a counting random variable N is valued in 0 1 2. Its
stochastic behaviour is characterized by the set of probabilities p k k= 0 1assigned
to the nonnegative integers, where p k = PrN = k. The (discrete) distribution of N associates
with each possible integer value k = 0 1 2 the probability p k that it will be the observed
value. The distribution must satisfy the two conditions:
p k ≥ 0 for all k and
+∑
k=0
p k = 1
i.e. the probabilities are all nonnegative real numbers lying between zero (impossibility) and
unity (certainty), and their sum must be unity because it is certain that one or other of the
values will be observed.
1.3.2 Probability Mass Function
In discrete distribution theory the p k s are regarded as values of a mathematical function, i.e.
p k = pk k = 0 1 2 (1.4)
where p· is a known function depending on a set of parameters . The function p·
defined in (1.4) is usually called the probability mass function. Different functional forms
lead to different discrete distributions. This is a parametric model.
The distribution function F N → 0 1 of N gives for any real threshold x, the probability
for N to be smaller than or equal to x. The distribution function F N of N is related to the
probability mass function via
x
∑
F N x = p k x∈ +
k=0
where p k is given by Expression (1.4) and where x denotes the largest integer n such that
n ≤ x (it is thus the integer part of x). Considering (1.4), F N also depends on .
1.3.3 Moments
There are various useful and important quantities associated with a probability distribution.
They may be used to summarize features of the distribution. The most familiar and widely
used are the moments, particularly the mean
EN =
+∑
k=0
kp k