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

Mixed Poisson Models for Claim Numbers 7
as possible, the proportion of times that an event A occurs would behave according to the
definition of Pr. Note that PrA is then the mathematical idealization of the proportion of
times A occurs.
1.2.5 Independent Events
Independence is a crucial concept in probability theory. It aims to formalize the intuitive
notion of ‘not influencing each other’ for random events: we would like to give a precise
meaning to the fact that the realization of an event does not decrease nor increase the
probability that the other event occurs. Formally, two events A and B are said to be
independent if the probability of their intersection equals the product of their respective
probabilities, that is, if PrA ∩ B = PrAPrB.
This definition is extended to more than two events as follows. The events in a family
of events are independent if for every finite sequence A 1 A 2 A k of events in ,
[ ]
⋂ k k∏
Pr A i = PrA i (1.1)
i=1
The concept of independence is very important in assigning probabilities to events. For
instance, if two or more events are regarded as being physically independent, in the sense
that the occurrence or nonoccurrence of some of them has no influence on the occurrence
or nonoccurrence of the others, then this condition is translated into mathematical terms
through the assignment of probabilities satisfying Equation (1.1).
i=1
1.2.6 Conditional Probability
Independence is the exception rather than the rule. In any given experiment, it is often
necessary to consider the probability of an event A when additional information about the
outcome of the experiment has been obtained from the occurrence of some other event B.
This corresponds to intuitive statements of the form ‘if B occurs then the probability of A is
p’, where B can be ‘March is rainy’ and A ‘the claim frequency in motor insurance increases
by 5 %’. This is called the conditional probability of A given B, and is formally defined as
follows. If PrB > 0 then the conditional probability PrAB of A given B is defined to be
PrAB =
PrA ∩ B
(1.2)
PrB
The definition of conditional probabilities through (1.2) is in line with empirical evidence.
Repeating a given experiment a large number of times, PrAB is the mathematical
idealization of the proportion of times A occurs in those experiments where B did occur,
hence the ratio (1.2).
It is easily seen that A and B are independent if, and only if,
PrAB = PrAB = PrA (1.3)