Math Is Fun Forum

getback0

Member

Registered: 2010-10-10

Posts: 25

Exponential Distribution Verbiage

I am not for sure if it is that I am misunderstanding the problem statement here or that the problem is just poorly stated -

================================================================================
An insurance policy reimburses dental expense, X, up to a maximum benefit of 250. The
probability density function for X is

f(x) =   ce^−0.004x for x >= 0,
           0 otherwise,

where c is a constant. Calculate the median benefit for this policy.
(A) 161 (B) 165 (C) 173 (D) 182 (E) 250

================================================================================

My question is -

When the question states the maximum benefit is 250 units, shouldnt the benefit density function be the value for f(x) only until 250 and 0 beyond?

I ask as this changes the answer. With benefit density limited to 250 (as the question states), my median calculation is around 94. Withouth the 250 unit limitation, the median works out to be 173 (Option - C) which matches the solution in my textbook.

Why is the solution not accounting for 250 unit limit? Any ideas?

THANKS IN ADVANCE BOBBYM!!!

- G

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Exponential Distribution Verbiage

Hi getback();

Thanks but I think you are overestimating me. Can I see what you did to get 94?

Did you get .004 for c?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

getback0

Member

Registered: 2010-10-10

Posts: 25

Re: Exponential Distribution Verbiage

if I assume, the density function goes all the way to infinity, then yes - I get c = 0.004 and the median is the solution of the equation

integral (0.004 e ^ -0.004*x ) over [0, x ] = 0.5  -works out to 173.4

However, since it mentioned maximum benefit of 250, to determine c, I used -

integral (pdf) over [0, 250] = 1 - this gives c of 0.00633

and then solving the same equation above for x gives 94.97

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Exponential Distribution Verbiage

Hi;

This problem appears on every actuary exam. See if these help you. These sites have some reasons why it must be .004.

http://docs.google.com/viewer?a=v&q=cac … dmqmMRl8kA

http://www.actuarialoutpost.com/actuari … 52708.html

http://lists.casact.org/read/messages?id=25584

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

getback0

Member

Registered: 2010-10-10

Posts: 25

Re: Exponential Distribution Verbiage

Got it  - makes sense -

1. I made a mistake assuming density function is 0 beyond x = 250- it is clearly 250

2. Clearly, if the median for the exponential density function works out to be less than the benefit limit, the limit doesnt change the median

Thanks bobbym! I got 2 more problems for you - Stay tuned!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Exponential Distribution Verbiage

Hi getback();

That is two chances to get one right. Or two chances to get two right, or...

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

getback0

Member

Registered: 2010-10-10

Posts: 25

Re: Exponential Distribution Verbiage

haha - I see the joke! thanks bobbym