Herschel-Maxwell demonstrates that for any linear combination of independent random variables where the probability density is a function of the distance from the origin (i.e., radially symmetric), the [[probability density function]] must be in the general form $f(r) = e^{Cr^2}$ where $C$ is some constant $C<0$ that allows us to use $e$ as the base. > [!Tip]- Additional Resources > -[Why pi is in the normal distribution](https://youtu.be/cy8r7WSuT1I?si=njLFdvz5tAhuViG8)| 3Blue1Brown.