- #1
mbigras
- 61
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Homework Statement
Show in detail that:
[tex]
\sigma_{x}^{2} = \int_{-\infty}^{\infty} (x -\bar{x})^{2} \frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{(x-X)^{2}}{2\sigma^{2}}} = \sigma^{2}
[/tex]
where,
[tex]
G_{X,\sigma}(x) = \frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{(x-X)^{2}}{2\sigma^{2}}}
[/tex]
Homework Equations
[tex]
\int u dv = uv -\int v du
[/tex]
The Attempt at a Solution
There are some hints that are given. Replace [itex]\bar{x}[/itex] with [itex]X[/itex] (according to the text, this is because they are equal after many trials). make the substitutions [itex]x-X=y[/itex] and [itex] y/\sigma=z[/itex]. Integrate by parts to obtain the result.
After following the hints I get the following integral:
[tex]
\frac{\sigma}{\sqrt{2\pi}} \int_{\infty}^{\infty} z^{2}e^{-\frac{z^{2}}{2}}dz
[/tex]
How would one go about using integration by parts for this integral?
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