Shadowdude
Cult of Personality
- Joined
- Sep 19, 2009
- Messages
- 12,146
- Gender
- Male
- HSC
- 2010
And I'm feeling quite happy about it because... it was long and arduous. It's not that hard in hindsight, I figure most HSC Extension 2 Maths students would be able to do it - I think it'd a pretty good test of their integration mastery.
I don't think there's anything abstractly difficult in it - it's just knowing what to do and being smart about it all.
So here it is:
![](https://latex.codecogs.com/png.latex?\bg_white \textup{Show that}\;E(X)=\alpha\beta\textup{, where:})
![](https://latex.codecogs.com/png.latex?\bg_white E(X)=\lim_{R \to \infty}\int_{0}^{R} x\frac{e^{-x/\beta}x^{\alpha - 1}}{\Gamma (\alpha) \beta^{\alpha}} dx)
You may also need the results:
![](https://latex.codecogs.com/png.latex?\bg_white \Gamma(\phi) = (\phi - 1)!, \;\;\phi = 0, 1, 2, 3,...)
![](https://latex.codecogs.com/png.latex?\bg_white \lim_{k \to \infty} k^{n}e^{-k/m} = 0, \textup{where k, m and n are real numbers})
Have fun!
EDIT: Also part of why I like this particular integral: "It looks difficult".
I don't think there's anything abstractly difficult in it - it's just knowing what to do and being smart about it all.
So here it is:
You may also need the results:
Have fun!
EDIT: Also part of why I like this particular integral: "It looks difficult".
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