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#1
02-04-2018, 03:25 PM
 JohnTravolski Member Non-Actuary Join Date: Aug 2016 Posts: 46
Moment Generating Function Technique

Let N be the number of claims made by a person. Assume that the number of claims varies with the type of person. Measure type of person by a second random variable J, which is nonnegative and follows some distribution F. Further, assume N conditioned on J=j follows a Poisson with mean j.
Find the distribution of N using the moment generating function technique; that is, express it as a function of the moment generating function of J.
#2
02-04-2018, 03:41 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,226

As it appears that j is lambda, integrate the conditional MGF of N (Poisson j). Look at the integral. It looks like the formula for a MGF with the dummy variable (usually called t) adjusted. That will give you the MGF of N.

Whether you can infer the distribution of N depends on the form of the MGF(the usual case being gamma j giving a negative binomial.
#3
02-04-2018, 03:51 PM
 JohnTravolski Member Non-Actuary Join Date: Aug 2016 Posts: 46

Integrate the MGF? I've never heard of integrating the MGF before; I don't understand why that works or what bounds I would be integrating over.

Did you mean that I should sum over the PMF instead? If not, then I'm still lost.
#4
02-04-2018, 03:56 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,226

I should probably say take the expected value of the conditional MFG. Look up the formula for the MGF of the Poisson. The parameter is in the exponent of an exponential.

In you case the parameter is a random variable with a known distribution. You know how to get the expected value of a random variable. This is just the expected value of a function of a random variable. The limits of the integral are the limits of the random variable (j).
#5
02-04-2018, 04:24 PM
 JohnTravolski Member Non-Actuary Join Date: Aug 2016 Posts: 46

Okay, I think I understand it now. So I'm using the iterated rule (smoothing technique / law of total expectation). I was confused originally since they specified that J was given as a constant, in which case the expected value of a constant is just another constant (so I wasn't getting anything useful). But if I just treat it as a random variable, then I get the following:

M_N(t)=M_J(e^t - 1)

Does that look right? (M_N is the MGF of N and M_J is the MGF of J)
#6
02-04-2018, 05:17 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,226

Quote:
 Originally Posted by JohnTravolski Okay, I think I understand it now. So I'm using the iterated rule (smoothing technique / law of total expectation). I was confused originally since they specified that J was given as a constant, in which case the expected value of a constant is just another constant (so I wasn't getting anything useful). But if I just treat it as a random variable, then I get the following: M_N(t)=M_J(e^t - 1) Does that look right? (M_N is the MGF of N and M_J is the MGF of J)
Yes
#7
02-04-2018, 05:25 PM
 JohnTravolski Member Non-Actuary Join Date: Aug 2016 Posts: 46

Thank you very much!

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