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A textbook I'm self-studying - Introduction to Mathematical Statistics by Hogg - has the following text:

T(a) = $\int_{0}^{\infty} y^{\alpha-1}e^{-y} dy$ [gamma function]

If $\alpha > 1$, an integration by parts shows that $T(\alpha) = (\alpha - 1)\int_{0}^{\infty} y^{\alpha-2}e^{-y} dy$.

I'm having trouble deriving this for myself. How does an Integration by Parts lead to this result?

Thank you.

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Let $dv=e^{-x}\ dx$ and $u=x^{\alpha-1}$. It thus follows that

$$v=-e^{-x},\quad du=(\alpha-1)x^{\alpha-2}\ dx$$

Thus,

$$\begin{align}\int x^{\alpha-1}e^{-x}\ dx&=\int u\ dv\\&=uv-\int v\ du\\&=-x^{\alpha-1}e^{-x}+(\alpha-1)\int x^{\alpha-2}e^{-x}\ dx\end{align}$$

And as the bounds go to $[0,\infty)$, we find that $uv\to0$, hence

$$\int_0^\infty x^{\alpha-1}e^{-x}\ dx=(\alpha-1)\int_0^\infty x^{\alpha-2}e^{-x}\ dx$$

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  • $\begingroup$ Thank you! This makes sense. For the bounds part of the proof: my understanding is that when the bounds approach zero, $-x^{\alpha - 1}$ is zero, making the $uv$ term zero. When the bounds approach $\infty$, $e^{-x}$ is zero, making the $uv$ term also zero. Is this correct? $\endgroup$
    – Daniel
    Mar 10 '17 at 19:55
  • $\begingroup$ @Daniel Particularly, as $x\to+\infty$, $e^{-x}$ goes to $0$ faster than the polynomial term goes to infinity. And I assume $\alpha>1$. $\endgroup$ Mar 10 '17 at 19:56
  • $\begingroup$ Thank you. How can one prove that $e^{-x}$ goes to 0 faster? $\endgroup$
    – Daniel
    Mar 10 '17 at 19:58
  • $\begingroup$ @Daniel L'Hospital's rule or similar methods. $\endgroup$ Mar 10 '17 at 19:59

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