Actuarial Outpost SOA 289 #15 / ASM 46.27
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 Short-Term Actuarial Math Old Exam C Forum

#1
05-05-2009, 07:50 PM
 ambersue SOA Join Date: Nov 2007 Location: CT Studying for FSA Modules College: University of South Carolina Alumni Favorite beer: Abita Purple Haze Posts: 6
SOA 289 #15 / ASM 46.27

Quote:
 You are given: The probability that an insured will have at least one loss during any year is p. The prior distribution for p is uniform on [0,0.5]. An insured is observed for 8 years and has at least one loss every year. Determine the posterior probability that the insured will have at least one loss during Year 9.
When answering this problem, I treated it as a Bernoulli/Beta conjugate with m=8 and k=8. I then solved for E[p|m,k] = a* / (a*+b*) = 9/10. However, since the prior distribution was uniform on [0,0.5] instead of [0,1], I adjusted my answer by 0.5. I arrived at the correct solution of 0.45, but I wanted to know if this was a fluke or if a similar adjustment would always work in cases where the prior is uniform on some interval [0,c]. Thoughts?
#2
05-05-2009, 09:11 PM
 badmaj5 Member CAS Join Date: Sep 2005 Location: Chicago, IL Posts: 454

I was under the impression that the beta/bernoulli trick only worked when the uniform was [0,1].
#3
05-05-2009, 10:31 PM
 Abraham Weishaus Member SOA AAA Join Date: Oct 2001 Posts: 7,193

Your trick works with 0.5 replaced by any other real number between 0 and 1 and "8" replaced by any other integer, and even with the uniform replaced by a beta distribution with b=1. Beyond that is harder for me to see.
#4
05-06-2009, 08:19 AM
 Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,688

My recollection is that for this to work you need to have all successes and no failures.

Jim Daniel
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#5
05-08-2009, 05:20 PM
 badmaj5 Member CAS Join Date: Sep 2005 Location: Chicago, IL Posts: 454

Quote:
 Originally Posted by Abraham Weishaus Your trick works with 0.5 replaced by any other real number between 0 and 1 and "8" replaced by any other integer, and even with the uniform replaced by a beta distribution with b=1. Beyond that is harder for me to see.
ASM Practice Exam 6, question 4:

The first line of the solution states "We cannot use the bernoulli/beta shortcut here since the uniform distribution is only up to .5"

I tried doing it using the shortcut and then just multiplying it by .5 at the end, and that gives me .375. What is different here?
#6
05-08-2009, 05:21 PM
 Abraham Weishaus Member SOA AAA Join Date: Oct 2001 Posts: 7,193

#7
05-08-2009, 05:26 PM
 badmaj5 Member CAS Join Date: Sep 2005 Location: Chicago, IL Posts: 454

So just to be sure I have it straight -

When we have uniform [0,1], the "regular" shortcut always works.

If we have uniform [0, b], and we observe 0 claims, then it would be (regular shortcut)*b.

But since in this problem, we observe 1 claim (not all successes), we must work it from first principles.

Is that accurate?
#8
04-19-2017, 12:44 PM
 davina SOA Join Date: Mar 2017 College: Alumni Posts: 4
Exam C SOA Sample #15

I am struggling with the last part of the solution to this Bayesian credibility problem. My understanding is that I can get the probability during the year 9 by integrating the posterior density function over 0 to 0.5. Why is the solution's integrand the posterior density multiply by p? Thank you.
#9
04-19-2017, 02:31 PM
 mattcarp Member CAS Join Date: Sep 2016 Studying for Exam 6 College: UC Berkeley Posts: 323

Quote:
 Originally Posted by davina I am struggling with the last part of the solution to this Bayesian credibility problem. My understanding is that I can get the probability during the year 9 by integrating the posterior density function over 0 to 0.5. Why is the solution's integrand the posterior density multiply by p? Thank you.
If you integrate the posterior density you will get 1. You need to integrate the posterior times the probability of what you want, given the parameter. Here we want the probability of one or more claims, which is given as p. Thus we integrate the posterior times p.
#10
04-19-2017, 03:10 PM
 davina SOA Join Date: Mar 2017 College: Alumni Posts: 4

Your explanation makes sense and helps, thanks.

 Tags asm, bayes, conjugate, soa