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#51
12-12-2006, 02:36 PM
 Nonpareil Note Contributor Join Date: Nov 2006 Location: Rocket City Studying for Exam C Posts: 833

Quote:
Originally Posted by colby2152
Quote:
 The first one that I was really confused with gave a joint m.g.f. You had to find a p.d.f. f(x). The other one gave a p.d.f. for lifetime of a 70 year old man. You had to find the probability that the man would live for the next five years, and then die within the next five I believe. The first part of that seemed easy, just integrate the pdf from zero to five. The second part was tough to figure out as I have never seen a problem designed like that. I decided to integrate the pdf to get the cdf and take a survivor function of that and evaluate from 5 to 10, and then multiply the two probabilities together. I did not get an answer that was given.
Did anyone get questions like these two?
These were not on my test, but for the first one, if you have a joint MGF M(s,t), I think you can find Mx(s) by setting t = 0.

Is the second one asking for a conditional probability? It doesn't seem that way from the description.
#52
12-12-2006, 04:05 PM
 colby2152 Note Contributor SOA Join Date: Feb 2006 Location: Virginia Studying for FAP College: PSU '07 Favorite beer: Oskar Blues Old Chub Scotch Ale Posts: 4,175

The second question stated... find the probaility that the 70 year old man lived for five years, and then died within the next five years.
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#53
12-12-2006, 04:10 PM
 ScubaChris Member CAS Join Date: May 2006 Location: Jersey City, NJ Studying for 6US College: TCNJ Favorite beer: Three Philosophers Posts: 524

Quote:
Originally Posted by mloiselle
Quote:
 Originally Posted by atomic if this is indeed the exact phrasing of the question.
No, that's the thing, it isn't the exact wording. The wording was such that I wasn't absolutely sure if the condition (X>400) applied. Did anyone actually get this question? The wording may have been "a claim was filed." Are claims filed if they are below the deductible?
I definitely had this question and am about 95% sure I put down 3/4.
#54
12-12-2006, 04:27 PM
 Nonpareil Note Contributor Join Date: Nov 2006 Location: Rocket City Studying for Exam C Posts: 833

Quote:
 Originally Posted by colby2152 The second question stated... find the probaility that the 70 year old man lived for five years, and then died within the next five years.
I would interpret that as Pr(5<X<10).
#55
12-12-2006, 04:42 PM
 ultrafilter Member Join Date: Nov 2006 Posts: 318

Quote:
 Originally Posted by christian Is the second one asking for a conditional probability? It doesn't seem that way from the description.
I don't remember the exact wording, but at the time it seemed pretty clear that they were asking for a conditional probability.
#56
12-12-2006, 04:44 PM
 atomic Member CAS Join Date: Jul 2006 Posts: 4,088

Quote:
Originally Posted by ultrafilter
Quote:
 Originally Posted by colby2152 Did anyone get questions like these two?
I got the second one. I set it up as P(X < 10 | X > 5), and got one of the answers given.
Life contingencies notation would be useful. Given the PDF for the lifetime random variable, we compute the survival function s(x), then calculate

$\,_{5|5}q_{70} = \frac{s(75)-s(80)}{s(70)}$.

So for example, if the PDF of age at death for a newborn life is given by f(x) = 1/100, x < 100 (DeMoivre), then the survival function is s(x) = 1 - x/100 and the desired probability is

$\,_{5|5}q_{70} = \frac{(1/4) - (1/5)}{3/10} = 1/6$.

But since the original PDF used in the question has not been mentioned, the above is just a sample calculation.
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#57
12-12-2006, 04:44 PM
 smythe Join Date: Dec 2006 Posts: 9

Yes, I had the question about the 70 year old man. Seemed very straight forward: Find Pr(5<x<10).
#58
12-12-2006, 05:04 PM
 Kazodev Member SOA Join Date: May 2004 Posts: 3,393

Quote:
Originally Posted by atomic
Quote:
 Originally Posted by ultrafilter I got the second one. I set it up as P(X < 10 | X > 5), and got one of the answers given.
Life contingencies notation would be useful. Given the PDF for the lifetime random variable, we compute the survival function s(x), then calculate

$\,_{5|5}q_{70} = \frac{s(75)-s(80)}{s(70)}$.

So for example, if the PDF of age at death for a newborn life is given by f(x) = 1/100, x < 100 (DeMoivre), then the survival function is s(x) = 1 - x/100 and the desired probability is

$\,_{5|5}q_{70} = \frac{(1/4) - (1/5)}{3/10} = 1/6$.

But since the original PDF used in the question has not been mentioned, the above is just a sample calculation.
isnt' that assuming that f is the pdf of a newborn? I was under the impression from the posts that f is the pdf of a person aged 70?
#59
12-12-2006, 05:11 PM
 mloiselle Member Join Date: Nov 2006 Posts: 86

Quote:
Originally Posted by atomic
Quote:
 Originally Posted by ultrafilter I got the second one. I set it up as P(X < 10 | X > 5), and got one of the answers given.
Life contingencies notation would be useful. Given the PDF for the lifetime random variable, we compute the survival function s(x), then calculate

$\,_{5|5}q_{70} = \frac{s(75)-s(80)}{s(70)}$.

So for example, if the PDF of age at death for a newborn life is given by f(x) = 1/100, x < 100 (DeMoivre), then the survival function is s(x) = 1 - x/100 and the desired probability is

$\,_{5|5}q_{70} = \frac{(1/4) - (1/5)}{3/10} = 1/6$.
But since the original PDF used in the question has not been mentioned, the above is just a sample calculation.
This result will be the same as:

If $f(x)$ is the pdf

$\qquad\qquad\qquad\qquad\frac{\int_{75}^{\infty} f(x) dx - \int_{80}^{\infty} f(x) dx}{\int_{70}^{\infty} f(x) dx} \qquad =\qquad \frac{\int_{75}^{80} f(x) dx}{\int_{70}^{\infty} f(x) dx$

Easily seen to be the probability that the lifespan is between 75 and 80 given that it is greater than 70. Could you refer me to a source to find out about the notation you use and why it is convenient. As a beginner, it seems bizarre.

Last edited by mloiselle; 12-12-2006 at 05:19 PM..
#60
12-12-2006, 05:19 PM
 Kazodev Member SOA Join Date: May 2004 Posts: 3,393

It's in chapter 3 of Bowers i.e. define t|u_q_x or something which means the probability of someone aged x dying between x+t and x+t+u. You learn it on course 3/M. So written in prob notation you have P(x+t < X < x+t+u | X > x) = [P(X < x + t + u) - P(X < x + t)]/P(X > x) = [S(x+t) - S(x+t+u)]/S(x)... I think, I haven't taken M

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