ASM, 5th ed, 12.10

[its_me]

The frequency of windstorms causing moe than 1,000,000 damage before inflation follows a NB distn with mean 0.2 & variance 0.4.
Uniform inflation of 5% affects the amount of damage
The severity of windstorms before inflation follows an exponential distn with mean 1,000,000
Calculate prob that there will be exactly 1 widstorn ina year causing moret han 2,000,000 damage after 1 year's inflation.

Answer 0.05383

I am getting 0.025204771

From mean & var of NB , i got r= 0.2 , beta=1
after 5% inflation, amt of damage follows expo(1050,000)
S(2,000,000) = 0.148858081
Now, we have NB(r =0.2 ,beta= 0.148858081)
So, prob of exactly 1 claim = 0.2*beta/(1+beta)^1.2 = 0.025204771

Solution is considering the relative prob of 2,000,000 to 1050,000 ,S(950,000)= exp(-19/21) to revise beta.WHY?

thanks!

[Gandalf]

The easiest way to see that there is something seriously wrong with your approach is to apply it to a much easier situation: 0 inflation instead of 5% inflation, and look at the distribution of hurricanes over 1,000,000.

As before, r=.2, beta = 1 for hurricanes over 1,000,000, from mean and variance.

Analogous to your use of S(2,000,000), with 0 inflation S(1,000,000) = .368

Analogous to your use of S(2,000,000), we have the distribution of hurricanes over 1,000,000 to be NB with r = .2, beta = .367.

No way. We are given that it is NB with r=.2, beta = 1.

Your problem is that you are adjusting the beta as if the frequency of hurricanes of any size in year 1 was NB(.2,1), but that is the frequency of hurricanes over 1,000,000. So when you adjust beta you must use the conditional distribution of hurricanes over 1,000,000 (or of over 1,050,000 after inflation.)