# parameter risk

Written by actuary on Tue Oct 21, 2008 12:43 am in blog actuary_math's blog under Actuary_math's blog -
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I believe that how few we consider the "parameter risk".

Suppose that the probability of occurring a accident is P (0<P<1).

For N polices, the probability of occurring n accidents is

N_C_n * P^n * (1-P)^(N-n)

(where N_C_n means the combinatorial number which is chosen n things form N things)

Suppose that the p.d.f.(probability density function) of the prior distribution is g(p),

f(p),the p.d.f. of the posterior distribution by Bayes' theorem

will be proportional to

P^n * (1-P)^(N-n) *g(p)

(because N_C_n is a constant which has no relations with p)

Now if the prior distribution is a uniform distribution, that is

g(p)=1(0<p<1)

f(p) is proportional to P^n*(1-P)^(N-n)

so

P follows to B(n+1,N+1)(beta distribution).

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