Actuarial Outpost SOA 289 #250
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#1
04-18-2009, 08:52 PM
 silvergrey Member Join Date: May 2008 Posts: 83
SOA 289 #250

can anybody show to me how come the mode is theata/2? I can not get it! Many thanks.
#2
04-18-2009, 09:24 PM
 daaaave David Revelle Join Date: Feb 2006 Posts: 2,972

The mode of an inverse exponential is one of the properties listed in the distribution tables that you get on the exam (on p. 6 of the version currently on the SOA website).
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#3
04-18-2009, 10:17 PM
 Actuarialsuck Member Join Date: Sep 2007 Posts: 6,114

Even if you didn't have that table, this is a simple exercise in calculus 1 right?

$f(x) \, = \, \frac{\theta e^{-\frac{\theta}{x}}}{x^2}$

Therefore using whichever differentiation rules you want, product, quotient, whatever, you get:

$f'(x) \, = \, \frac{\theta^2}{x^4} e^{-\frac{\theta}{x}} \, - \, \frac{2\theta}{x^3} e^{-\frac{\theta}{x}}$

Simple factoring gives us

$f'(x) \, = \, \frac{\theta}{x^3} e^{-\frac{\theta}{x}} \left[ \frac{\theta}{x} \, - \, 2 \right]$

As calculus rules go, you set it = 0, convince yourself why the first term = 0 is not what you want (do you see why?), therefore the 2nd term must be i.e.

$\frac{\theta}{x} \, - \, 2 \, = \, 0 \, \Rightarrow \, x \, = \, \frac{\theta}{2}$
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#4
11-18-2015, 05:08 PM
 ScottKelly Member SOA Join Date: Jul 2012 Posts: 305
The difference between SOA sample #152 and #250

On Sample #152 when we calculated the likelihood function, for the densities we used, we used the conditional density of being greater than the deductible of 500. We are not given information about losses that aren't at least meeting the deductible.

On Sample #250 when we calculated the likelihood function, for the densities we used for losses above the deductible, we used the unconditional densities. In this question, we were given information about losses that did not meet the deductible.

Is the reason that in #250 we did not have to condition losses above the deductible as being above the deductible, because we were given information about the losses that did not meet the deductible? If we were not given information about the losses that did not meet the deductible, would it have been okay to use conditional densities then for losses that were above the deductible?
#5
11-18-2015, 05:33 PM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 30,885

Not just OK to use the conditional, necessary to use the conditional
#6
06-13-2018, 06:39 PM
 laxtuary CAS SOA Join Date: Nov 2017 College: PSU '14 Posts: 1
Losses Below Deductible

I'm still a bit confused on what clues us in to why we don't need to condition on being above the deductible in this problem? It seems like we have just as much information given about losses below the deductible as we would on any other problem like this.
#7
06-13-2018, 07:10 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 7,931

There are 11 losses. You know seven are below the deductible, you just don't know the specific values. If you were just given the losses above the deductible you would use the conditional.
#8
06-13-2018, 08:09 PM
 Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,643

Quote:
 Originally Posted by ScottKelly On Sample #152 when we calculated the likelihood function, for the densities we used, we used the conditional density of being greater than the deductible of 500. We are not given information about losses that aren't at least meeting the deductible. On Sample #250 when we calculated the likelihood function, for the densities we used for losses above the deductible, we used the unconditional densities. In this question, we were given information about losses that did not meet the deductible. Is the reason that in #250 we did not have to condition losses above the deductible as being above the deductible, because we were given information about the losses that did not meet the deductible? If we were not given information about the losses that did not meet the deductible, would it have been okay to use conditional densities then for losses that were above the deductible?
Where does #250 say anything about a deductible?
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