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#1
05-09-2018, 05:21 PM
 loco_pabs CAS Join Date: Jul 2014 College: Wartburg College Posts: 4
#210

Each life within a group medical expense policy has loss amounts which follow a compound Poisson process with $\lambda=0.16$. Given a loss, the probability that it is for Disease 1 is 1/16.

Loss amount distributions have the following parameters:
Code:
                              Mean per loss             Std. Dev. per loss
--------------------------------------------------------------------------
Disease 1                         5                              50
Other Diseases                   10                              20
Premiums for a group of 100 independent lives are set at a level such that the probability (using normal approximation to the distribution for aggregate losses) that aggregate losses for the group will exceed aggregate premiums for the group is 0.24.

A vaccine which will eliminate Disease 1 and costs 0.15 per person has been discovered.

Define:
A = the aggregate premium assuming that no one obtains the vaccine, and
B = the aggregate premium assuming that everyone obtains the vaccine and the cost of the vaccine is a covered loss.

Calculate A/B.
-----------------------------------------------------------------------------------------

Why are we able to multiply 0.16 ($\lambda$) by each of the probabilities for Disease 1 or Other diseases?
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#2
05-22-2018, 09:27 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 7,638

Quote:
 Originally Posted by loco_pabs Each life within a group medical expense policy has loss amounts which follow a compound Poisson process with $\lambda=0.16$. Given a loss, the probability that it is for Disease 1 is 1/16. Loss amount distributions have the following parameters: Code:  Mean per loss Std. Dev. per loss -------------------------------------------------------------------------- Disease 1 5 50 Other Diseases 10 20 Premiums for a group of 100 independent lives are set at a level such that the probability (using normal approximation to the distribution for aggregate losses) that aggregate losses for the group will exceed aggregate premiums for the group is 0.24. A vaccine which will eliminate Disease 1 and costs 0.15 per person has been discovered. Define: A = the aggregate premium assuming that no one obtains the vaccine, and B = the aggregate premium assuming that everyone obtains the vaccine and the cost of the vaccine is a covered loss. Calculate A/B. ----------------------------------------------------------------------------------------- Why are we able to multiply 0.16 ($\lambda$) by each of the probabilities for Disease 1 or Other diseases?
The Poisson has the property that if the outcomes fall into different classes with known probabilities, then the number in each class will have its own Poisson distribution.
#3
05-22-2018, 11:58 PM
 ARodOmaha Member SOA Join Date: May 2016 Location: Omaha, NE Studying for MFE College: University of Nebraska (alma mater) Favorite beer: Captain Morgan Posts: 106

Quote:
 Originally Posted by Academic Actuary The Poisson has the property that if the outcomes fall into different classes with known probabilities, then the number in each class will have its own Poisson distribution.
Yes, that is why the problem states that it is a "compound Poisson", i.e. it can be separated into proportional Poisson's. Conversely, you can add Poisson's together. (I learned this after getting that exact problem wrong.)
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