Let XXX have the characteristic function ϕX(t)=e−∣t∣\phi_X(t) = e^{-|t|}ϕX(t)=e−∣t∣. What is the PDF f(x)f(x)f(x)?
f(x)=1π(1+x2)f(x) = \frac{1}{\pi(1+x^2)}f(x)=π(1+x2)1
f(x)=12e−∣x∣f(x) = \frac{1}{2}e^{-|x|}f(x)=21e−∣x∣
f(x)=2π(1+x2)2f(x) = \frac{2}{\pi(1+x^2)^2}f(x)=π(1+x2)22
f(x)=12πe−x2/2f(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}f(x)=2π1e−x2/2