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Bama Gambler
08-12-2002, 02:38 PM
First of all I changed my mind about Mahler's notes. Previously I claimed Mahler's notes were overrated. I made this statement without really giving them a chance. I guess I was overwhelmed by the number of pages and all the footnotes. After using the notes for over 2 weeks I have a new opinion: Mahler's notes for course 3 are AWESOME!! I hope to purchase his course 4 notes soon (i.e. I hope to pass course 3 this fall). Ok, now that I have cleared that up...

Stochastic Models Section 7 - Stationary Distributions of Markov Chains
Problems #12
Why is there a factor of 2 in the computation of the second moment?

Thanks in advance for any replies,
Bama Gambler

burton leon reynolds
08-12-2002, 04:22 PM
I don't have the problem in front of me, but a factor of two can be introduced into the second moment through a summation by parts(I think). This may or maynot apply here.

retaker
08-12-2002, 05:09 PM
Just be glad they took the continuous time markov chains off the syllabus.

Howard Mahler
08-13-2002, 12:37 PM
The Exponential Distribution has a second moment of 2(mean^2).
Other distributions have different formulas.
See the Tables attached to the exam or
"Mahler's Guide to Loss Distributions."

Howard Mahler

P.S. The questions numbers differ between editions of my study guides, so it would help if you mentioned the edition.

MathGuy
08-13-2002, 01:32 PM
Just because I want to take a brief break from work:

Remember, when taking an integral from 0 to inf of t<sup>a-1</sup>e<sup>-t</sup> dt, you get http://physics.nist.gov/Images/Gamma.gif(a), also known as (a-1)! = (a-1)*(a-2)*...(2)(1), assuming a is an integer.

To calculate the kth moment of an exponential distribution with parameter http://physics.nist.gov/Images/theta.gif, calculate the integral from 0 to infinity of f(x)*x<sup>k</sup> dx, or:

Int( f(x)*x<sup>k</sup>, dx)

= Int( x<sup>k</sup>e<sup>(-x/http://physics.nist.gov/Images/theta.gif)</sup>/http://physics.nist.gov/Images/theta.gif, dx)

For some substitution, let y = x/http://physics.nist.gov/Images/theta.gif. Then x = http://physics.nist.gov/Images/theta.gify, and dx = http://physics.nist.gov/Images/theta.gifdy. So, our integral is now:

= Int( (http://physics.nist.gov/Images/theta.gify)<sup>k</sup>e<sup>-y</sup>/http://physics.nist.gov/Images/theta.gif, http://physics.nist.gov/Images/theta.gifdy)
= Int( http://physics.nist.gov/Images/theta.gif<sup>k</sup>y<sup>k</sup>e<sup>-y</sup>, dy)
= http://physics.nist.gov/Images/theta.gif<sup>k</sup> * Int( y<sup>k</sup>e<sup>-y</sup>, dy)
= http://physics.nist.gov/Images/theta.gif<sup>k</sup> * http://physics.nist.gov/Images/Gamma.gif(k-1)
= http://physics.nist.gov/Images/theta.gif<sup>k</sup>k!

In this specific case, k = 2, so the second moment is http://physics.nist.gov/Images/theta.gif<sup>2</sup>2! = 2http://physics.nist.gov/Images/theta.gif<sup>2</sup>.

As usual, I'm very, very sorry.

Bama Gambler
08-13-2002, 01:54 PM
Thanks!! Can't believe I missed that! :oops: