RE: May, 1997 #40

Livingston, Erik F. ( (no email) )
Wed, 13 Oct 1999 08:11:10 -0500

This one I got, Not by understanding what discrete convolutions is but by
using the discrete probabilities maybe it is the same thing, I didn't look
up convolutions.
The key to me for this problem was noticing the type of claim is not random
one of each is always chosen. So I only need to look at each case and the
probabilities of the different states.
So I went through manually and exhausted combinations.
For example choose the 5,000 BI serious claim (State A .4) Which
combinations of the 2 other claims go above 7,500
I am going to shift into some abbreviation here for the rest of the note
BIn and PD.
With BI serious 5,000 the combinations that push the total over 7,500 are
(BIn - st.A, PD- st B -probability .4*.4)
(BIn - st B, PD - all states Probability .4*1) >7,500
(BIn - st C, PD - all states probability .2*1) > 7,500
So if BI serious 5,000 is the claim (probability .4)
The chances of going over 7,500 are (.4*.4+.4*1+.2*1) = .76
Total chance then is .4 * .76 (BI serious * other comb) where > 7,500

I did this process for BI serious of 3,000 (probability of .4)
And got only one over 7,500
(BIn st B , PD st B) probility .4*.4 = .16

For BI serious of 2,000 there is no combination that goes over 7,500 prob 0%

So the final answer is weighting the above cases based on each groups
probability or sum of (BI serious prob * remaining combination prob) for
each BI serious state
..4*.76 + .4 * .16 +.2 * 0= .368
This also doesn't match your actex answer. But I it matches the CAS exam
solution (from the ordered past exams).

-----Original Message-----
From: Daslcnt@aol.com [SMTP:Daslcnt@aol.com]
Sent: Tuesday, October 12, 1999 7:29 PM
To: studygroup5a@lists.casact.org
Subject: May, 1997 #40

Can anyone tell me how they would (or ifyou should) solve this
problem using
discrete convolutions? I did not get what the Actex solutions came
up with.
The problem is this:

An insurance company writes only automobile insurance. Depending on
the
initial claim information received by the company, preliminary
opening
reserves are established by the claims department in the following
manner:

Type of Claim State A State B State C
Bodily Injury (serious) $5,000 $3,000 $2,000
Bodily Injury (not ser.) $1,000 $3,000 $2,000
Property Damage $1,000 $2,000 $1,000

According to the cmopany's statistics, 60% of claims are property
damage, 30%
are non-serious BI, and 10% are serious BI. Also, the company's
claims are
distributed 40% State A, 40% State B, and 20% State C.

Each claim file includes the details for only one of the three types
of claim
every month; the claims manager does an audit of each examiner's
files to
ensure that the initial reserves are properly set by pulling three
files at
random--one for each type of loss.

What is the probability that the total dollars of reserves involved
in any
examiners audit exceeds $7,500?

I set it up as follows:
S f(x1) f(x2) f(x3) f(x1x2) f(s)
$1000 0 .4 .6 0
$2000 .2 .2 .4 .08
$3000 .4 .4 0
$4000 0 0 0
$5000 .4 0 0
$6000 0 0 0
$7000 0 0 0
$8000 0 0 0

ETC.....Then I did 1- pr(s<7500) = 7.6%

Actex got .32; but I think they made a mistake.......HELP!!