Actuarial Outpost Actex Problem Set 5 #13
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#1
04-28-2012, 02:15 PM
 oceankyle Member Non-Actuary Join Date: Dec 2011 Posts: 154
Actex Problem Set 5 #13

This problem is pretty simple, but I'm a little confused on one of the concepts. I'm guessing its something simple also but...

A company insures homes in three cities J, K, L. Since sufficient distance separates the cities, it is reasonable to assume that the losses occurring in these cities are independent. The moment generating functions for the loss distributions of the cities are .... (insert the MGFs here)

Let X represent the combined losses of the 3 cities. Calculate E[X^3]

The solution entails multiplying the 3 moment generating functions together than taking the third derivative and substituting 0 for t...

Ok, So my question is why are we multiplying the moment generating functions when we are looking for the combined loss? Shouldn't the solution involve the sum of the 3 expected values?

Also on page 145 it explains that you can find a moment generating function of a new variable by taking the weighted average of the moment generating functions of the variables you are combining.

I just don't quite understand the concept/difference here. thanks for any help.
#2
04-28-2012, 03:32 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 7,662

Quote:
 Originally Posted by oceankyle This problem is pretty simple, but I'm a little confused on one of the concepts. I'm guessing its something simple also but... It reads: A company insures homes in three cities J, K, L. Since sufficient distance separates the cities, it is reasonable to assume that the losses occurring in these cities are independent. The moment generating functions for the loss distributions of the cities are .... (insert the MGFs here) Let X represent the combined losses of the 3 cities. Calculate E[X^3] The solution entails multiplying the 3 moment generating functions together than taking the third derivative and substituting 0 for t... Ok, So my question is why are we multiplying the moment generating functions when we are looking for the combined loss? Shouldn't the solution involve the sum of the 3 expected values? Also on page 145 it explains that you can find a moment generating function of a new variable by taking the weighted average of the moment generating functions of the variables you are combining. I just don't quite understand the concept/difference here. thanks for any help.
The third moment of the combined loss is E[(X1+X2+X3)^3] so you will have all the cross product terms. If you only wanted the mean then you could sum.
If the pdf is a "mix" or weighted average then the MGF will be a weighted average but the weights have to sum to 1. You have a mix when there is more than one distribution generating losses and you are picking one at random. An example is a population comprised of a number of risk classes where you are picking someone from the population at random. With a sum you would be picking one from each class and summing their losses.
#3
04-28-2012, 04:39 PM
 oceankyle Member Non-Actuary Join Date: Dec 2011 Posts: 154

Quote:
 Originally Posted by Academic Actuary The third moment of the combined loss is E[(X1+X2+X3)^3] so you will have all the cross product terms. If you only wanted the mean then you could sum. If the pdf is a "mix" or weighted average then the MGF will be a weighted average but the weights have to sum to 1. You have a mix when there is more than one distribution generating losses and you are picking one at random. An example is a population comprised of a number of risk classes where you are picking someone from the population at random. With a sum you would be picking one from each class and summing their losses.
Ah that clears it up some... Could you take the third derivative of each moment generating function add them, and then divide by 3?

Why did we take the product of the moment generating functions when looking for a sum?

Thanks again.
#4
04-28-2012, 09:50 PM
 Actuarialsuck Member Join Date: Sep 2007 Posts: 6,111

Quote:
 Originally Posted by oceankyle Ah that clears it up some... Could you take the third derivative of each moment generating function add them, and then divide by 3? Why did we take the product of the moment generating functions when looking for a sum? Thanks again.
Because e^{x+y} = e^x e^y
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