Results 1 to 3 of 3

Math Help - Taylor expansion?

  1. #1
    Senior Member
    Joined
    Jul 2006
    Posts
    364

    Taylor expansion?

    How did the authors expand \phi(\omega) below? (Related to the characteristic function derived from the moment generating function)

    I'm self studying this, and not sure what area of math covers these topics? I know how to take a Taylor expansion of a typical function, but what about this improper integral? Or the sum? I'm not sure what they did.



    EDIT: \phi_X(\omega)=M_X(i\omega)=E(e^{i\omega X})

    Ah, nevermind, I figured it out -- they used several theorems relating to expectation and started from the E(e^{i\omega X}) instead of the integral or sum.

    But how do they know the integral converges? What test did they use?
    Last edited by scorpion007; June 27th 2010 at 12:28 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,545
    Thanks
    1394
    The integral (and series) converge only for some functions f(x). Here, I assume that f(x) is some probability density function and so \sum f(x)= 1 or \int f(x)= 1dx. In that case, you can use the "comparison test" for both sum and integral, comparing e^{e\omega x}f(x) to f(x).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Jul 2006
    Posts
    364
    Thanks a lot!

    So because exp(iwx)f(x) <= f(x) on (-infty .. infty) and int {f(x)} converges, so does the former function.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Taylor Expansion
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 6th 2010, 12:15 PM
  2. taylor expansion of erf(x)
    Posted in the Calculus Forum
    Replies: 3
    Last Post: January 10th 2010, 10:26 AM
  3. Taylor Expansion x^2+y^2+z^2=9
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 28th 2009, 05:26 PM
  4. taylor expansion
    Posted in the Calculus Forum
    Replies: 2
    Last Post: February 5th 2009, 04:59 PM
  5. Taylor Expansion....
    Posted in the Calculus Forum
    Replies: 4
    Last Post: November 13th 2007, 08:23 PM

Search Tags


/mathhelpforum @mathhelpforum