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Exponential Density Function

  1. Apr 8, 2013 #1
    1. The problem statement, all variables and given/known data

    Given f(x;λ) = [itex]cx^{2}e^{-λx}[/itex] for x ≥ 0

    Determine what c must be (as a function of λ) then determine the maximum likelihood estimator of λ.

    3. The attempt at a solution

    So i'm supposed to integrate this from 0 to infinity, from what i can gather.

    Let u = [itex]x^{2}[/itex], du = 2xdx, dv = [itex]e^{-λx}[/itex] and v = [itex]-e^{-λx} / λ[/itex]

    After a bit of work i end up with:

    -c/λ [ [itex]x^{2}e^{-λx}|_{0}^{∞} + 2( xe^{-λx}/λ |^{∞}_{0})[/itex] ]

    What throws me off is that evaluating this leaves me with -c/λ( 0 ), which has to be wrong...
  2. jcsd
  3. Apr 8, 2013 #2


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    You have to integrate the second term from 0 to infinity, not just evaluate it. You'll need to integrate by parts again.
  4. Apr 8, 2013 #3

    Ray Vickson

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    It might be easier to recognize that
    [tex] x e^{-\lambda x} = - \frac{\partial}{\partial \lambda} e^{- \lambda x}, [/tex]
    and so forth.
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