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

14 Actuarial Modelling of Claim Counts
Binomial Distribution
The Binomial distribution describes the outcome of a sequence of n independent Bernoulli
trials, each with the same probability q of success. The probability that success is the outcome
in exactly k of the trials is
( n
pkn q = q
k)
k 1 − q n−k k= 0 1n (1.9)
and 0 otherwise. Formula (1.9) defines the Binomial distribution. There are now two
parameters: the number of trials n (also called the exponent, or size) and the success
probability q. Henceforth, we write N ∼ inn q to indicate that N is Binomially
distributed, with size n and success probability q.
Moments of the Binomial Distribution
The mean of N ∼ inn q is
EN =
n∑
k=1
= nq
n!
k − 1!n − k! qk 1 − q n−k
n∑
PrM = k − 1 = nq (1.10)
where M ∼ inn − 1q. Furthermore, with M as defined before,
so that the variance is
EN 2 =
n∑
k=1
= nq
k=1
n!
k − 1!n − k! kqk 1 − q n−k
n∑
k PrM = k − 1
k=1
= nn − 1q 2 + nq
VN = EN 2 − nq 2 = nq1 − q (1.11)
We immediately observe that the Binomial distribution is underdispersed, i.e. its variance is
smaller than its mean : VN = nq1 − q ≤ EN = nq.
Probability Generating Function and Closure under Convolution for the
Binomial Distribution
The probability generating function of N ∼ inn q is
( ) n∑ n
N z = qz k 1 − q n−k = 1 − q + qz n (1.12)
k=0
k
Note that Expression (1.12) is the Bernoulli probability generating function (1.8), raised to
the nth power. This was expected since the Binomial random variable N can be seen as the