Actuarial Outpost Integrating exponentials
 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

US LIFE, ANNUITY AND INVESTMENT JOBS

#1
05-09-2011, 09:08 AM
 Chocolate Muffins And Pie Member Join Date: Jan 2008 Posts: 223
Integrating exponentials

I keep getting frustrated with problems requiring integration of an exponential. This has been helpful, feel free to point out other ways or poke holes in this.

To integrate x^a*e^-bx

1) write out -e^-bx* ( )
2) in the parentheses, start with x^a and add a term for each derivative until you reach a constant (so x^a + ax^(a-1) +a(a-1)x^(a-2).....)
3) for each of those terms, add a denominator starting with b^1 and increasing the exponent until you reach b^(a+1)
4) evaluate at the limits of the integral.
#2
05-09-2011, 10:24 PM
 asile Member Join Date: Jun 2008 Posts: 103

Quote:
 Originally Posted by Chocolate Muffins And Pie I keep getting frustrated with problems requiring integration of an exponential. This has been helpful, feel free to point out other ways or poke holes in this. To integrate x^a*e^-bx 1) write out -e^-bx* ( ) 2) in the parentheses, start with x^a and add a term for each derivative until you reach a constant (so x^a + ax^(a-1) +a(a-1)x^(a-2).....) 3) for each of those terms, add a denominator starting with b^1 and increasing the exponent until you reach b^(a+1) 4) evaluate at the limits of the integral.
Why not just use the fact that this is a constant away from being a gamma distribution? It integrates to 1 / constant if you are going from 0 to infinity.

Otherwise, it might be faster to back out the constant and use the gamma -> poisson trick for an integer n/alpha parameter.

 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 01:31 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