Weird Mathematics

Today, we are going to do some weird mathematics.

Weird Mathematics 1

If you have taken advanced statistics, you will know this identity.

${\displaystyle E\left(x\right)^{2}+\sigma ^{2}=E\left(x^{2}\right)}$

We are going to let ${\displaystyle f=E\left(x\right)}$, and let ${\displaystyle g=E\left(x^{2}\right)}$.

This effectively boils down to: ${\displaystyle f^{2}+\sigma ^{2}=g}$

Solving for the standard deviation gives us:

$${\displaystyle \sigma ={\sqrt {g-f^{2}}}}$$If you favor the integral version, we have:

$${\displaystyle \sigma ={\sqrt {\left({\frac {1}{b-a}}\right)\int _{a}^{b}f\left(y\right)^{2}dy-\left(\left({\frac {1}{b-a}}\right)\int _{a}^{b}f\left(y\right)dy\right)^{2}}}}$$Continuing to level 3, we have:

$${\displaystyle {\frac {\sqrt {\frac {\left({\frac {1}{b-a}}\right)\int _{a}^{b}f\left(y\right)^{3}dy-\left(\left({\frac {1}{b-a}}\right)\int _{a}^{b}f\left(y\right)dy\right)^{3}}{\left(\left({\frac {1}{b-a}}\right)\int _{a}^{b}f\left(y\right)dy\right)}}}{\sqrt {3}}}}$$