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Simplify the following formula with Gamma functions

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[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?
 
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Answers and Replies

CompuChip
Science Advisor
Homework Helper
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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.
 
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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).
 
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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
 
360
21
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...
 
tiny-tim
Science Advisor
Homework Helper
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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
 
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21
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".
 
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