Thursday, April 28, 2016

Variance Solids: Volumes and $\mathscr{L}^2$, Part 2

Last time, we talked about how the classical $\mathscr{L}^2$ norm of a (one-variable) function could be visualized as a solid consisting of squares. Of course, there's another formula that is defined by almost the same formula, with an extra factor of $\pi$: the solid of revolution of the function about the $x$-axis: \[
V = \int_a^b \pi f(x)^2\,dx
It looks pretty cool, although not quite as cool as twisted square cross-sections.

I wanted to mention another common application of $\mathscr{L}^2$ norms that probably is more familiar than any other: the statistical concept of variance. For functions $f$ defined on a continuous domain, we can define an average value:\[
\overline{f} = \frac1{b-a}\int_a^b f(x)\,dx.
\] which basically is the height of a rectangle over the same domain $[a,b]$ that has the same area as the area under $f$. For example, for \[
f(x) = \sin \left( x \right)\sin \left( t \right)+\frac{1}{3}\sin \left( 5x \right)\sin \left( 5t \right)+\frac{1}{3}\sin \left( 10x \right)\sin \left( 10t \right)+\frac{1}{5}\sin \left( 15x \right)\sin \left( 15t \right)
(choosing $t=3.030303$), it looks like:

(the blue line in the middle is the mean value). The variance $\sigma^2$ is then a measure of how far $f$ is from its mean:\[ \sigma^2 = \frac{1}{b-a}\int_a^b (f(x) - \overline{f})^2\,dx. \] This of course can be visualized exactly as last time, namely, make a bunch of square cross sections about the line $y=\overline{f}$, and possibly do funny things to such squares, such as rotate it. But ... for kicks, let's add a factor of $\pi(b-a)$:\[ \pi(b-a)\sigma^2 = \int_a^b\pi (f(x) - \overline{f})^2\,dx, \] which is precisely the same quantity as the solid obtained by revolving $f(x)$ around its mean $y=\overline{f}$. The resulting solid is the title picture.

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