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
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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.
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