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Statistics Gamma Distribution question

  1. Oct 6, 2011 #1
    1. The problem statement, all variables and given/known data
    Let X have a gamma distribution with parameters α and β.
    Show that P(X ≥ 2αβ) ≤ (2/e)2

    2. Relevant equations

    f(x) = pfd of a Gamma

    3. The attempt at a solution

    I began by solving for P(X ≥ 2αβ) by doing ∫ f(x) dx from 2αβ to ∞
    I set y=x/β for substitution.
    and I got up to 1/[itex]\Gammaα[/itex] * ∫yα-1 e-y dy from 2αβ to ∞

    I don't really know what to do from here.
    I know that ∫yα-1 e-y dy from 0 to ∞ = [itex]\Gammaα[/itex]
    but I'm kind of lost because of the 2αβ

    I am not even sure if this approach to solving this problem is correct.

    For the second part of the question, we have a theorem that says :
    P(X≥ α) ≤ eαtM(t)

    My guess for the second part is to simply solve for eαtM(t)
    Is that correct?

    Any help will be greatly appreciated. Thank you.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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