# magic of the Gaussian distribution

29 Jan 2014The width $\sigma$ of a Gaussian distribution \(p(x)= \frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{x^2}{2\sigma^2}\right)\) can be obtained by integration of the second moment, \(\langle x^2\rangle = \int\mathrm{d}x\:p(x)x^2 = \sigma^2\) or by differentiation and solving \(\mathrm{d}^2p/\mathrm{d}x^2=0\). Can you show that this property is unique for the Gaussian distribution?