Simplify the following formula with Gamma functions

[montoyas7940]

\frac{\beta^\alpha \Gamma(\alpha + 1)}{\Gamma (\alpha) \beta^{\alpha+ 1}}

= \frac{\alpha \Gamma (\alpha)}{ \beta \Gamma (\alpha)}

= \frac{\alpha}{ \beta}

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?

 

[CompuChip]

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

 

[montoyas7940]

Thanks, CompuChip.
As it turns out there is more to this that I was not given. \Gamma(\alpha) is a function. \Gamma(n) = (n-1)! (factorial).

 

[happyg1]

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

 

[montoyas7940]

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]

… use wikipedia … !

Hi montoyas7940;!

\Gamma(\alpha + 1)}\,=\alpha\Gamma(\alpha)\,.​

See: http://en.wikipedia.org/wiki/Gamma_function#Properties

 

[happyg1]

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

 

[montoyas7940]

So,

\frac{\beta^\alpha \alpha\Gamma(\alpha)}{\beta^{\alpha+ 1}\Gamma (\alpha) }

= \frac{\alpha \Gamma (\alpha)}{ \beta \Gamma (\alpha)}

= \frac{\alpha}{ \beta}

I like it! Maybe I will "survive".