Deriving the MGF for the Weibull Distribution

  1. I'm attempting to derive the MGF for the Weibull distribution. I know that E([tex]e^{tx}[/tex]), which equals the integral shown here:

    http://www.wolframalpha.com/input/?i=Integrate%5Be^%28t*x%29*%28k%2F%CE%BB%29*%28x%2F%CE%BB%29^%28k-1%29*e^-%28x%2F%CE%BB%29^k%2Cx%5D

    where the parameters are k and λ.

    The answer is found here:

    http://www.wolframalpha.com/input/?i=Sum%5B%28t^n+%CE%BB^n%29%2Fn%21%2C+{n%2C+0%2C+Infinity}%5D*gamma%281%2B%281%2Fk%29%29

    So I see that I need to get the gamma function and the series representation for e^(t*λ) to show up in order to get the right answer. I've been trying to use a change of variable such as u = (x/λ)^k, and I feel like I've been getting close, but can't exactly get it right. Can someone guide me along with this? Thank you.

    *For some reason it keeps putting a space in the URL, so just take them out
     
  2. jcsd
  3. Stephen Tashi

    Stephen Tashi 4,409
    Science Advisor
    2014 Award

    Those links you gave might be temporary URLs. I didn't get the first one to work.
     
  4. In the first url try copy and pasting the whole thing, but taking out the space between the 2 and the F

    Similarly, for the second url take out the space between the I and the nfinity. If this doesn't work let me know and I'll attempt to repost what I'm trying to show in another format.
     
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