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Show the gamma density function integrates to 1

  1. Jun 29, 2013 #1
    1. The problem statement, all variables and given/known data
    Show the gamma density function integrates to 1.


    2. Relevant equations
    Assume α > 0, λ > 0, t > 0
    g(t) = [itex]\frac{λ^α}{\Gamma (α)} t^{α-1}e^{-λt}[/itex]
    [itex]\Gamma (α)= \int_0^∞ t^{α-1} e^{-t} dt [/itex]
    3. The attempt at a solution

    Show [itex]\int_0^∞ \frac{λ^α}{\Gamma (α)} t^{α-1}e^{-λt} dt = 1[/itex]

    [itex]\int_0^∞ \frac{λ^α}{\Gamma (α)} t^{α-1}e^{-λt} dt[/itex]
    = [itex]\frac{λ^α}{\Gamma (α)} \int_0^∞ t^{α-1}e^{-λt}dt[/itex]
    = [itex]\frac{λ^α}{\Gamma (α)} \int_0^∞ t^{α-1}e^{-t}e^λdt[/itex]
    = [itex]\frac{λ^αe^λ}{\Gamma (α)} \int_0^∞ t^{α-1}e^{-t}dt[/itex]
    = [itex]\frac{λ^αe^λ \Gamma (α)}{\Gamma (α)}[/itex]
    = [itex]λ^αe^λ[/itex]

    ...where did I lose the plot?
     
  2. jcsd
  3. Jun 29, 2013 #2

    Dick

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    Science Advisor
    Homework Helper

    You lost it when you said ##e^{-λt}=e^{-t}e^λ##. That's not true. I would try the variable substitution ##u=λt##.
     
    Last edited: Jun 29, 2013
  4. Jul 1, 2013 #3
    Looks like I need to refresh myself on the laws of exponents.

    That substitution did the trick, thanks.
     
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