Actuarial Outpost SOA 165
 Register Blogs Wiki FAQ Calendar Search Today's Posts Mark Forums Read
 FlashChat Actuarial Discussion Preliminary Exams CAS/SOA Exams Cyberchat Around the World Suggestions

#1
05-09-2008, 02:52 AM
 Doc Holiday Member Join Date: Aug 2006 Location: Tombstone Favorite beer: Anything that defiles myself Posts: 1,004
SOA 165

I dont get how E[X ^ 3] is calculated. Some help please?
Is this easy and I'm just burnt out, or am I screwed?
__________________
The key to happiness is low expectations.
#2
05-09-2008, 03:12 AM
 ElDucky Free Mason Join Date: Jul 2004 Location: In a van, down by the river Studying for Let me worry about blank Favorite beer: Trappistes Rochefort 8 Posts: 39,755

There's another one later on that is similar. I don't get it either. I didn't see much about in the manual but I am having another look.
#3
05-09-2008, 04:21 AM
 jraven Member Join Date: Aug 2007 Location: New Hampshire Studying for nothing! College: Penn State Posts: 1,304

$X \wedge 3 = 3 - (3 - X)_+$ (just draw graph of y = 3 and y = x ^ 3)

But $E[3] = 3$, and since $(3 - X)_+ = 0$ for $X \geq 3$ we can compute the expectation as
$E[(3-X)_+] = (3-0) f_S(0) + (3-1) f_S(1) + (3-2) f_S(2)$

which is one way of seeing that
$E[X \wedge 3] = 3 - [3 f_S(0) + 2 f_S(1) + 1 f_S(2)]$

[Conceptually, the only difference between X ^ 3 and 3 occurs when x = 0, 1 or 2 and so you just need to subtract off the right stuff to account for that.]
__________________
The Poisson distribution wasn't named after a fish -- it was named after a man ... who was named after a fish.
#4
05-09-2008, 11:14 AM
 Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,580

It's easy to compute E[S ^ 3] directly. The possible values of S are 0, 1, 2, 3, 4, 5, ... and so the corresponding possible values of S ^ 3 are 0, 1, 2, 3, 3, 3, ... . So

$E[S\wedge 3]=f_S(0)0+f_S(1)1+f_S(2)2+[1-f_S(0)-f_S(1)-f_S(2)]3$

Jim Daniel
__________________
Jim Daniel
Jim Daniel's Actuarial Seminars
www.actuarialseminars.com
jimdaniel@actuarialseminars.com
#5
05-09-2008, 01:33 PM
 Doc Holiday Member Join Date: Aug 2006 Location: Tombstone Favorite beer: Anything that defiles myself Posts: 1,004

Quote:
 Originally Posted by Jim Daniel It's easy to compute E[S ^ 3] directly. The possible values of S are 0, 1, 2, 3, 4, 5, ... and so the corresponding possible values of S ^ 3 are 0, 1, 2, 3, 3, 3, ... . So $E[S\wedge 3]=f_S(0)0+f_S(1)1+f_S(2)2+[1-f_S(0)-f_S(1)-f_S(2)]3$ Jim Daniel
This makes sense. Thanks!
__________________
The key to happiness is low expectations.
#6
10-01-2008, 09:10 AM
 bmathew22 Member Join Date: Jan 2006 Posts: 126

Ok, so now I understand how to do this calculation once I have computed the fs', but how were the fs' computed?
#7
10-01-2008, 09:48 AM
 Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,580

Just examine how S can eual ech of those values, and then compute the probability for each case.

Jim Daniel
__________________
Jim Daniel
Jim Daniel's Actuarial Seminars
www.actuarialseminars.com
jimdaniel@actuarialseminars.com
#8
10-01-2008, 06:31 PM
 bmathew22 Member Join Date: Jan 2006 Posts: 126

Ok, so fs(2) would be one loss of 2 or 2 losses of 1. Makes sense, thanks!
#9
09-09-2016, 12:11 AM
 umich Member SOA AAA Join Date: Mar 2016 Location: Detroit, MI College: University of Michigan Alum Posts: 131

Quote:
 Originally Posted by Jim Daniel It's easy to compute E[S ^ 3] directly. The possible values of S are 0, 1, 2, 3, 4, 5, ... and so the corresponding possible values of S ^ 3 are 0, 1, 2, 3, 3, 3, ... . So $E[S\wedge 3]=f_S(0)0+f_S(1)1+f_S(2)2+[1-f_S(0)-f_S(1)-f_S(2)]3$ Jim Daniel
Jim, I'm having trouble deciding whether to put condition to the expectation or not. I was able to calculate E(S-3)+ but I then divided it by Pr (S>3). Sometimes I can get it right but other times I use the wrong one. So why is it wrong to use the conditional expectation? Thank you!
__________________
P FM MLC C MFE FAP APC ILA-LP
#10
09-09-2016, 02:07 AM
 Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,580

Quote:
 Originally Posted by umich Jim, I'm having trouble deciding whether to put condition to the expectation or not. I was able to calculate E(S-3)+ but I then divided it by Pr (S>3). Sometimes I can get it right but other times I use the wrong one. So why is it wrong to use the conditional expectation? Thank you!
I don't understand why you would even THINK about conditioning on something. Does the expression E[X ^ 3] or E[(X - 3)+] have a "conditioned on" symbol | ? I certainly don't see one. You ask why is it wrong to use conditional expectation. I instead ask where you see anything that indicates that you SHOULD use conditional expectation. E[X ^ 3] simply asks you to compute the average value of the smaller of X and 3. No conditioning, none, nada, niet, nyet, nee.
__________________
Jim Daniel
Jim Daniel's Actuarial Seminars
www.actuarialseminars.com
jimdaniel@actuarialseminars.com

 Thread Tools Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off

All times are GMT -4. The time now is 01:51 PM.

 -- Default Style - Fluid Width ---- Default Style - Fixed Width ---- Old Default Style ---- Easy on the eyes ---- Smooth Darkness ---- Chestnut ---- Apple-ish Style ---- If Apples were blue ---- If Apples were green ---- If Apples were purple ---- Halloween 2007 ---- B&W ---- Halloween ---- AO Christmas Theme ---- Turkey Day Theme ---- AO 2007 beta ---- 4th Of July Contact Us - Actuarial Outpost - Archive - Privacy Statement - Top