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

8 Actuarial Modelling of Claim Counts
Note that this interpretation of independence is much more intuitive than the definition given
above: indeed the identity expresses the natural idea that the realization or not of B does not
increase nor decrease the probability that A occurs.
1.2.7 Random Variables and Random Vectors
Often, actuaries are not interested in an experiment itself but rather in some consequences of
its random outcome. For instance, they are more concerned with the amounts the insurance
company will have to pay than with the particular circumstances which give rise to the
claims. Such consequences, when real-valued, may be thought of as functions mapping
into the real line .
Such functions are called random variables provided they satisfy certain desirable
properties, precisely stated in the following definition: A random variable X is a measurable
function mapping to the real numbers, i.e., X→ is such that X −1 − x ∈
for any x ∈ , where X −1 − x = ∈ X ≤ x. In other words, the measurability
condition X −1 − x ∈ ensures that the actuary can make statements like ‘X is less
than or equal to x’ and quantify their likelihood. Random variables are mathematical
formalizations of random outcomes given by numerical values. An example of a random
variable is the amount of a claim associated with the occurrence of an automobile accident.
A random vector X = X 1 X 2 X n T is a collection of n univariate random variables,
X 1 , X 2 , , X n , say, defined on the same probability space Pr. Random vectors are
denoted by bold capital letters.
1.2.8 Distribution Functions
In many cases, neither the universe nor the function X need to be given explicitly.
The practitioner has only to know the probability law governing X or, in other words, its
distribution. This means that he is interested in the probabilities that X takes values in
appropriate subsets of the real line (mainly intervals).
To each random variable X is associated a function F X called the distribution function
of X, describing the stochastic behaviour of X. Of course, F X does not indicate what is the
actual outcome of X, but shows how the possible values for X are distributed (hence its
name). More precisely, the distribution function of the random variable X, denoted as F X ,is
defined as
F X x = PrX −1 − x ≡ PrX ≤ x x ∈
In other words, F X x represents the probability that the random variable X assumes a
value that is less than or equal to x. IfX is the total amount of claims generated by some
policyholder, F X x is the probability that this policyholder produces a total claim amount
of at most E x. The distribution function F X corresponds to an estimated physical probability
distribution or a well-chosen subjective probability distribution.
Any distribution function F has the following properties: (i) F is nondecreasing, i.e.
Fx ≤ Fy if x