Actuarial Outpost MLE question involving both frequency and severity distributions
 Register Blogs Wiki FAQ Calendar Search Today's Posts Mark Forums Read
 FlashChat Actuarial Discussion Preliminary Exams CAS/SOA Exams Cyberchat Around the World Suggestions

 DW SimpsonActuarial JobsVisit our site for the most up to date jobs for actuaries. Actuarial Salary SurveysProperty & Casualty, Health, Life, Pension and Non-Tradtional Jobs. Actuarial Meeting ScheduleBrowse this year's meetings and which recruiters will attend. Contact DW SimpsonHave a question? Let's talk. You'll be glad you did.

 Short-Term Actuarial Math Old Exam C Forum

#1
09-24-2017, 04:29 PM
 Seasplash Member CAS SOA Join Date: Jun 2014 Location: Arlington, VA Studying for MLC College: Cal Poly San Luis Obispo Posts: 307
MLE question involving both frequency and severity distributions

There are two claims in 1 year, and the frequency distribution is Poisson with parameter theta. Those two claims are x = 3 and x = 6 and the severity follows an Inverse exponential distribution with parameter theta. Estimate theta using MLE.

Would the answer be 8/3? I just want to make sure I'm doing it correctly. If someone can show the solution that would be great too, thanks.
__________________
P FM MFE C MLC
#2
09-25-2017, 03:30 PM
 catdynasty528 Member Join Date: Jul 2011 College: Ball State University Posts: 73

I also got 8/3.

I'm guessing your solution is correct if you want to post it and the community can verify.
#3
09-25-2017, 04:00 PM
 ALivelySedative Member CAS Join Date: Dec 2013 Location: Land of the Pine College: UNC-Chapel Hill Alum Favorite beer: Red Oak Posts: 2,774

How bad is it that I already don't remember how to do this off the top of my head...??
__________________
Stuff | 6 | ACAS | FCAS stuff
#4
09-25-2017, 10:29 PM
 Milton Friedman CAS Join Date: Mar 2015 College: JMU Posts: 12

that's the answer I got as well-I suppose that's right.
#5
09-25-2017, 11:01 PM
 Seasplash Member CAS SOA Join Date: Jun 2014 Location: Arlington, VA Studying for MLC College: Cal Poly San Luis Obispo Posts: 307

Alright thanks! Just needed the confirmation!
__________________
P FM MFE C MLC
#6
12-06-2017, 10:16 AM
 Bash Member SOA Join Date: Jun 2010 Posts: 62

Can someone please post the solution. I don't know why I'm getting 4/3 and not 8/3 as gotten by everybody.
__________________
P FM MFE C MLC
VEE: Statistics Corporate Finance Economics
#7
12-06-2017, 10:42 AM
 daaaave David Revelle Join Date: Feb 2006 Posts: 3,005

$L(\theta) = \Pr[N=2] \cdot f(3 \mid \theta) \cdot f(6 \mid \theta) = e^{-\theta} \cdot \frac{\theta^2}{2} \cdot \frac{\theta e^{-\theta/3}}{9} \cdot \frac{\theta e^{-\theta/6}}{36} = c \theta^4 e^{-3\theta/2}$
$\ell(\theta) = \ln(c) + 4 \ln(\theta) - \frac{3\theta}{2}$
$\ell^\prime(\theta) = 0 + \frac{4}{\theta} - \frac{3}{2}$
and solving gives theta-hat = 8/3
__________________