Prove that

\[\int_{-\infty}^{\infty} f^{2}(x) d x\]

equals \(\frac{1}{2} \pi^{-1 / 2}, 1, \frac{1}{4}, \frac{1}{6}\), and \(\frac{2}{3}\) for the \(\mathrm{N}(0,1), \operatorname{Uniform}\left(-\frac{1}{2}, \frac{1}{2}ight)\), LaPlace \((0,1)\), Logistic \((0,1)\), and Triangular \((-1,1,0)\) densities, respectively.