Actuarial Outpost
 
Go Back   Actuarial Outpost > Exams - Please Limit Discussion to Exam-Related Topics > SoA/CAS Preliminary Exams > Short-Term Actuarial Math
FlashChat Actuarial Discussion Preliminary Exams CAS/SOA Exams Cyberchat Around the World Suggestions

DW Simpson Global Actuarial & Analytics Recruitment
Download our 2017 Actuarial Salary Survey
now with state-by-state salary information!


Short-Term Actuarial Math Old Exam C Forum

Reply
 
Thread Tools Search this Thread Display Modes
  #1  
Old 06-15-2018, 04:31 PM
RockOn RockOn is offline
Member
SOA
 
Join Date: Dec 2013
Studying for Exam C
Favorite beer: Polygamy Porter
Posts: 80
Default SOA #304

Hello!
Can someone please explain to me how we know that the sum of the alphas add to 1?

Thank you.

I'm having trouble posting an image of the problem, so sorry.
__________________
EXAMS: P | FM | MFE | MLC | STAM | PA |
VEE: ECON | STATS | CORP FINANCE |
FAP: MOD 1| MOD 2| MOD 3| MOD 4| MOD 5| IA| MOD 6| MOD 7| MOD 8| FA |
Reply With Quote
  #2  
Old 06-15-2018, 04:57 PM
Academic Actuary Academic Actuary is offline
Member
 
Join Date: Sep 2009
Posts: 8,226
Default

In the limit as y goes to infinity, all the cdfs go to one.
Reply With Quote
  #3  
Old 06-15-2018, 05:55 PM
RockOn RockOn is offline
Member
SOA
 
Join Date: Dec 2013
Studying for Exam C
Favorite beer: Polygamy Porter
Posts: 80
Default

Quote:
Originally Posted by Academic Actuary View Post
In the limit as y goes to infinity, all the cdfs go to one.
Thank you for the response!
Yes, I understand that the cdfs sum to 1.

What confuses me is that the density function associated with each alpha is different, ie, each exponential density has a different theta. I can see how the alphas would sum to 1 if the density function associated with each alpha used the same theta but they don't.??
__________________
EXAMS: P | FM | MFE | MLC | STAM | PA |
VEE: ECON | STATS | CORP FINANCE |
FAP: MOD 1| MOD 2| MOD 3| MOD 4| MOD 5| IA| MOD 6| MOD 7| MOD 8| FA |
Reply With Quote
  #4  
Old 06-15-2018, 07:31 PM
Academic Actuary Academic Actuary is offline
Member
 
Join Date: Sep 2009
Posts: 8,226
Default

Quote:
Originally Posted by RockOn View Post
Thank you for the response!
Yes, I understand that the cdfs sum to 1.
The cdfs don't sum to 1. The cdfs weighted by the alphas sum to 1. If the alphas didn't sum to 1 then the cdf of the mixed distribution would not be 1 in the limit.
Reply With Quote
  #5  
Old 06-15-2018, 07:45 PM
RockOn RockOn is offline
Member
SOA
 
Join Date: Dec 2013
Studying for Exam C
Favorite beer: Polygamy Porter
Posts: 80
Default

Quote:
Originally Posted by Academic Actuary View Post
The cdfs don't sum to 1. The cdfs weighted by the alphas sum to 1. If the alphas didn't sum to 1 then the cdf of the mixed distribution would not be 1 in the limit.

Right. Still confused.

Say that:

alpha 1 = .1
alpha 2 = .2
alpha 3 = .3
alpha 4 = .4

Each one of these alphas are being multiplied by a different density function (i.e. and exponential with different thetas). How does the fact that the alphas sum to one ensure that the cdf will be one?

I'm so confused.
__________________
EXAMS: P | FM | MFE | MLC | STAM | PA |
VEE: ECON | STATS | CORP FINANCE |
FAP: MOD 1| MOD 2| MOD 3| MOD 4| MOD 5| IA| MOD 6| MOD 7| MOD 8| FA |
Reply With Quote
  #6  
Old 06-15-2018, 09:00 PM
Academic Actuary Academic Actuary is offline
Member
 
Join Date: Sep 2009
Posts: 8,226
Default

Quote:
Originally Posted by RockOn View Post
Right. Still confused.

Say that:

alpha 1 = .1
alpha 2 = .2
alpha 3 = .3
alpha 4 = .4

Each one of these alphas are being multiplied by a different density function (i.e. and exponential with different thetas). How does the fact that the alphas sum to one ensure that the cdf will be one?

I'm so confused.
Each unweighted density integrates to 1. Each weighted density integrates to the alpha weight. The sum of the weighted densities will integrate to 1.
Reply With Quote
  #7  
Old 06-15-2018, 09:01 PM
daaaave daaaave is offline
David Revelle
 
Join Date: Feb 2006
Posts: 3,000
Default

Y is a mixture of 4 different distributions, namely exponentials with means theta_1, theta_2, theta_3, and theta_4 respectively. The alpha_i are probabilities of selecting each of those 4 different components of the mixture, and as such sum to 1 because the sum of the probabilities is 1.
__________________

Follow us on Twitter, Facebook, and LinkedIn
Reply With Quote
  #8  
Old 06-17-2018, 06:18 PM
RockOn RockOn is offline
Member
SOA
 
Join Date: Dec 2013
Studying for Exam C
Favorite beer: Polygamy Porter
Posts: 80
Default

Hello and thank you for all the responses. They've all been very helpful and I think I have a much better understanding of this problem. Your explanations all make sense.
I hope that I can quickly assimilate this type of problem should it come up on an exam.

Again, thank you all.
__________________
EXAMS: P | FM | MFE | MLC | STAM | PA |
VEE: ECON | STATS | CORP FINANCE |
FAP: MOD 1| MOD 2| MOD 3| MOD 4| MOD 5| IA| MOD 6| MOD 7| MOD 8| FA |
Reply With Quote
Reply

Thread Tools Search this Thread
Search this Thread:

Advanced Search
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off


All times are GMT -4. The time now is 08:22 PM.


Powered by vBulletin®
Copyright ©2000 - 2018, Jelsoft Enterprises Ltd.
*PLEASE NOTE: Posts are not checked for accuracy, and do not
represent the views of the Actuarial Outpost or its sponsors.
Page generated in 0.18331 seconds with 9 queries