Actuarial Outpost Darth Vader Rule Question
 User Name Remember Me? Password
 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

 Thread Tools Search this Thread Display Modes
#1
05-05-2018, 02:24 AM
 fjmvasa CAS SOA Join Date: Apr 2018 Posts: 2
Darth Vader Rule Question

Hi everyone! I would like to ask if the Darth Vader also applies for x < 0? Since there is a rule for a < x < infinity.
#2
05-05-2018, 03:35 AM
 Michael Mastroianni SOA Join Date: Jan 2018 Posts: 12

If $X$ is a random variable defined on $[a,\infty)$ and $a<0$ then $X-a$ is defined on $[0,\infty)$ right? So $X-a$ is nonnegative and you can use the rule:

$\displaystyle E[X-a]=\int_{0}^{\infty} [1-F_{X-a}(t)] \, dt=\int_{0}^{\infty} [1-P(X-a\leq t)] \, dt=\int_{0}^{\infty} [1-P(X\leq t+a)] \, dt=\int_{a}^{\infty} [1-P(X\leq x)] \, dx$

Where the last part uses substitution: $x=t+a$

But also from linearity we know $\displaystyle \int_{a}^{\infty} [1-F_{X}(x)] \, dx=E[X-a]=E[X]-a$

Add $a$ to both sides for: $\displaystyle E[X]=a+\int_{a}^{\infty} [1-F_{X}(x)] \, dx$

So in this limited case it does extend to negative values. I hope that helps.
__________________
Michael Mastroianni, ASA
Video Course for Exam 1/P: www.probabilityexam.com
#3
05-06-2018, 10:03 PM
 fjmvasa CAS SOA Join Date: Apr 2018 Posts: 2

Thank you very much Michael!

 Thread Tools Search this Thread Search this Thread: Advanced Search 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 08:55 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

Powered by vBulletin®
Copyright ©2000 - 2018, Jelsoft Enterprises Ltd.
*PLEASE NOTE: Posts are not checked for accuracy, and do not
represent the views of the Actuarial Outpost or its sponsors.
Page generated in 0.25924 seconds with 11 queries