매우 중요하다
Suppose that \(\mu\) is a probability measure with probability distribution function \(F\) and characteristic function \(\phi\). If \[ \int_{-\infty}^\infty |\phi(t)|dt<\infty \] then \(\mu\) has a bounded, uniformly continuous density given by \[ f(x)=F'(x)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{-itx}\phi(t)dt. \]
Standard normal distribution의 characteristic function은 \(\phi(t)=e^{-t^2/2}\)이다. 당연히 이는 integrable function이다. \((\because \int_{-\infty}^\infty e^{-t^2/2}dt= \sqrt{2\pi}<\infty)\). 때문에, \[ f(x)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{-itx}e^{-\frac{t^2}{2}}dt= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{i(-x)t}\frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}}dt= \frac{1}{\sqrt{2\pi}}E(e^{-ixT})=\frac{1}{\sqrt{2\pi}}\phi(-x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}. \]