I have the following past exam question:
A portfolio has 100 risks with identical and independent number of claims. The number of claims for one risk has a Poisson distribution with mean lambda. The prior distribution is Gamma(alpha=4, theta=1/50). During year 1, 90 risks had 0 claims, 7 had 1 claim, 2 had 2 claims, and 1 had 3 claims. Determine the Buhlmann estimates of the expected number of claims for the portfolio in year 2.
So here's what I did...it is wrong somewhere, but where?
mu=E[E[N|Lambda]]=2/25
v=E[Var[N|Lambda]]=2/25
a=Var[E[N|Lambda]]=1/625
Z=1/(1+v/a)=1/(1+50)=1/51
X(sample average)=(90*0+7*1+2*2+1*3)/100=0.14
Thus the Buhlmann estimate should be = ZX+(1-Z)mu=(1/51)(0.14)+(50/51)(2/25) = 0.081176
But the answer should be 12? Where did I go wrong? Thanks
A portfolio has 100 risks with identical and independent number of claims. The number of claims for one risk has a Poisson distribution with mean lambda. The prior distribution is Gamma(alpha=4, theta=1/50). During year 1, 90 risks had 0 claims, 7 had 1 claim, 2 had 2 claims, and 1 had 3 claims. Determine the Buhlmann estimates of the expected number of claims for the portfolio in year 2.
So here's what I did...it is wrong somewhere, but where?
mu=E[E[N|Lambda]]=2/25
v=E[Var[N|Lambda]]=2/25
a=Var[E[N|Lambda]]=1/625
Z=1/(1+v/a)=1/(1+50)=1/51
X(sample average)=(90*0+7*1+2*2+1*3)/100=0.14
Thus the Buhlmann estimate should be = ZX+(1-Z)mu=(1/51)(0.14)+(50/51)(2/25) = 0.081176
But the answer should be 12? Where did I go wrong? Thanks
Buhlmann Model Question
