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#1




Confusing question about SUM vs MIX of Poisson Process
I am completely confused between SUM and MIX of poisson process.
Problem 1 ( ASM page 820 example 53C) Good insureds submit claims at a poisson rate of 0.2 per year. Bad insureds submit claims at a poisson rate of 0.8 per year. 75% of the insureds are good. For a single insured whom you cannot classify as good or bad, calculate the probability of at least 2 claims submitted in 3 years. The answer treats it as a mixture poisson process. Problem 2 ( ASM page 824 question 53.6) Insurance policies are sold to 30 good drivers and 20 bad drivers. For a good driver, the annual number of claims has a poisson distribution with mean 0.2;For a bad driver, the annual number of claims has a poisson distribution with mean 0.4. Calculate the probability of at least one claim in one month from this group. The answer treats it as a sum of Poisson Process. I don't see difference between the above two problems. Why problem 1 is a mixture of poisson process, while problem 2 is a sum of poisson process? Can anyone help me out? Thanks... 
#2




You can easily tell whether a random variable is a sum: Can you identify random variables X_1, X_2, ... , X_n which when summed up will generate the random variable you're studying? You can do that with #2 (X_1 through X_30 will be the good drivers and X_31 to X_50 the bad ones), but how could you do that with #1? You can't. The number of claims from that insured is X, but X cannot be expressed as a sum of other random variables of interest.

#3




I know this is an older post, but the question is quite valid. Abe's answer is also very insightful.
I can see another difference. The first one asks for something related to one individual, we don't know to which group it belongs. ("For a single insured whom you cannot classify as good or bad, calculate the probability of at least 2 claims submitted in 3 years."). So we find out each conditional probability for each group, and use the law of total probability to figure out the expected value. The second one is asking for something related to the group ("Calculate the probability of at least one claim in one month from this group."). So we look at the group's equivalent Poisson, with parameter equal to the sum of the individual Poisson parameters. For all of these (yearly to monthly and yearly to triennial) we also use a nice property of the Poisson akin to the sum of parameters. I wonder if this is correct. If not please let me know.
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#4




In case 2 you are certain as to the composition of the group while in case 1 you are not. In case 1 there are 2 urns and you are drawing an insured from one at random. In case 2 you have 2 urns and you are drawing a fixed number of insureds from each urn.
By the way this material is no longer on LTAM and I don't think it's directly on STAM. It would be on MAS I. 
#5




Thank you AA, I did a search and it just showed up... I didn't realize it was LTAM... I am going for STAM, after that I will try LTAM. It would be a subject for STAM, I believe, as we had similar cases.
Thank you for the great input.
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German ______________ Prelims: VEE: 
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