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Integration involving Gamma Distribution

  1. Oct 27, 2009 #1
    1. The problem statement, all variables and given/known data
    Evaluate [tex]
    \int^{\infty}_0 \frac{1}{B^{\alpha} \Gamma(\alpha)} \theta^{\alpha + x -1}e^{- \theta(1+B)/B} d \theta

    2. Relevant equations

    3. The attempt at a solution

    \int^{\infty}_0 \frac{1}{B^{\alpha} \Gamma(\alpha)} \theta^{\alpha + x -1}e^{- \theta(1+B)/B}d \theta

    = \frac{1}{B^{\alpha} \Gamma(\alpha)} \int^{\infty}_0 \theta^{\alpha + x -1}e^{- \theta(1+B)/B}d \theta

    Let [tex]u = \theta(1+B)/B[/tex] and [tex]du = \frac{1 + B}{B} d\theta[/tex]

    = \frac{1}{B^{\alpha} \Gamma(\alpha)} \int^{\infty}_0 \frac{1 + B}{B} \frac{Bu}{1+B}^{\alpha + x -1}e^{- u} du

    I'm stuck here. Not sure what to do next.
  2. jcsd
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