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Homework Help: Simplify the following formula with Gamma functions

  1. Mar 30, 2008 #1
    [tex]\frac{\beta^\alpha \Gamma(\alpha + 1)}{\Gamma (\alpha) \beta^{\alpha+ 1}}[/tex]

    = [tex]\frac{\alpha \Gamma (\alpha)}{ \beta \Gamma (\alpha)}[/tex]

    = [tex]\frac{\alpha}{ \beta}[/tex]

    This is the solution. In trying to get the middle expression out of the first I quickly end up with a mess. How should I approach this?
     
    Last edited: Mar 30, 2008
  2. jcsd
  3. Mar 30, 2008 #2

    CompuChip

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    Please try to use LaTeX tags, or write the fractions on one line using brackets. So either write
    [tex]\frac{B^a C(a + 1)}{C(a) B^(a + 1)}[/tex]
    or write
    B^a C(a + 1) / [ C(a) B^(a+1) ]

    Also, what is C(a)? Is it C multiplied by a? Or is C a function and is C(a) the function value in a? Or did you forget a caret and did you mean C^a ? In the last case I get C/B which is closest to the supposed solution you gave.
     
  4. Mar 30, 2008 #3
    Thanks, CompuChip.
    As it turns out there is more to this that I was not given. [tex]\Gamma(\alpha)[/tex] is a function. [tex]\Gamma(n)[/tex] = (n-1)! (factorial).
     
  5. Mar 30, 2008 #4
    Yes, yes, the lovely Gamma function. Is this for a statistics class?
    Did you solve it after you found out about the Gamma function?
    CC
     
  6. Mar 30, 2008 #5
    Probability for risk management, Happyg1. Not a class though, just for kicks.
    I haven't solved it yet, I had to walk away for a while...
     
  7. Mar 30, 2008 #6

    tiny-tim

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  8. Mar 30, 2008 #7
    We just got finished studying the Gamma Distribution in Mathematical Stats class. It's an interesting distribution. We went through the entire derivation of the properties that tiny-tim is showing you. Very cool...and kinda morbid. Our Professor explained that the Gamma distribution is the distribution used for "life testing"...the waiting time until death. Now you know about a function that models the waiting time until a "success" (which is death) occurs.
    I love math!
    CC
     
  9. Mar 30, 2008 #8
    So,

    [tex]\frac{\beta^\alpha \alpha\Gamma(\alpha)}{\beta^{\alpha+ 1}\Gamma (\alpha) }[/tex]

    = [tex]\frac{\alpha \Gamma (\alpha)}{ \beta \Gamma (\alpha)}[/tex]

    = [tex]\frac{\alpha}{ \beta}[/tex]

    I like it! Maybe I will "survive".
     
    Last edited: Mar 30, 2008
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