Actuarial Outpost SOA #131
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#1
05-22-2018, 11:42 AM
 RockOn Member SOA Join Date: Dec 2013 Studying for Exam C Favorite beer: Polygamy Porter Posts: 53
SOA #131

Hello!

I think I understand this problem for the most part. Just need a little help in understanding how the following formula was derived (in SOA solution):

(-1/lambda)*ln(1-u)

Is this supposed to be the inverse of the poisson pgf?

Thank you!
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#2
05-22-2018, 11:57 AM
 Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,617

Quote:
 Originally Posted by RockOn Hello! I think I understand this problem for the most part. Just need a little help in understanding how the following formula was derived (in SOA solution): (-1/lambda)*ln(1-u) Is this supposed to be the inverse of the poisson pgf? Thank you!
No. When frequency of events is Poisson with mean lambda, the "times" (or fractions of an exposure unit) between events are mutually independent Exponential random variables with mean theta = 1 / lambda. Also, given a value U of a Uniform random variable on (0,1), the inversion-simulated value of an Exponential is - theta ln ( 1 - U ).

Technically, #131 asks about a Poisson _Process_, and Poisson Processes are no longer on the syllabus. However, it could have just said that you were simulating a Poisson random variable and then it would be on the syllabus.
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Jim Daniel
Jim Daniel's Actuarial Seminars
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#3
05-22-2018, 11:57 AM
 RockOn Member SOA Join Date: Dec 2013 Studying for Exam C Favorite beer: Polygamy Porter Posts: 53

Quote:
 Originally Posted by RockOn Hello! I think I understand this problem for the most part. Just need a little help in understanding how the following formula was derived (in SOA solution): (-1/lambda)*ln(1-u) Is this supposed to be the inverse of the poisson pgf? Thank you!
...I meant to say: Is this supposed to be the inverse of the poisson CDF?
__________________
EXAMS: P | FM | MFE | MLC | C | 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 |
#4
05-22-2018, 12:00 PM
 Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,617

Quote:
 Originally Posted by RockOn ...I meant to say: Is this supposed to be the inverse of the poisson CDF?
__________________
Jim Daniel
Jim Daniel's Actuarial Seminars
www.actuarialseminars.com
jimdaniel@actuarialseminars.com
#5
05-22-2018, 12:53 PM
 RockOn Member SOA Join Date: Dec 2013 Studying for Exam C Favorite beer: Polygamy Porter Posts: 53

Quote:
 Originally Posted by Jim Daniel Again, no. See my answer to your first post on #131.
Thank you! I hadn't seen your first post when I posted my second post.

Thank you again!
__________________
EXAMS: P | FM | MFE | MLC | C | 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 |