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

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Actuarial Modelling of Claim Counts
( n
k)
=
( )
n! n
k!n − k! =
n − k
Note that the binomial coefficient is sometimes denoted as Cn k , especially in the Frenchwritten
mathematical literature, but here we adhere to the more standard notation ( n
k)
. The
Gamma function · is defined as
x =
∫
0
t x−1 exp−tdt
x > 0
As for any positive integer n, we have n = n−1!, the Gamma function can be considered
as an interpolation of the factorials defined for positive integers. Integration by parts shows
that x + 1 = xx for any positive real x. When a and b are positive real numbers, the
definition of the binomial coefficient is extended to positive integers as
( a a + 1
=
b)
a − b + 1b + 1
The incomplete Gamma function · · is defined as
t = 1
t
∫
0
x t−1 exp−xdx t ≥ 0
A real-valued random variable is denoted by a capital letter, for instance X. The
mathematical expectation operator is denoted as E·. For instance, EX is the expectation
of the random variable X. The variance is VX, given by VX = EX 2 − EX 2 .A
random vector is denoted by a bold capital letter, for instance X = X 1 X n T . Matrices
should not be confused with random vectors (the context will make this clear). The variancecovariance
matrix of X has covariances CX i X j = EX i X j − EX i EX j outside the
main diagonal (that is, for i ≠ j) and the variances VX i along the main diagonal.
The probability distributions used in this book are summarized next:
• the Bernoulli distribution with parameter 0