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Question from ECE QE August 2000

a) The Laplacian density function is given by $ f\left(x\right)=\frac{A}{2}e^{-A\left|x\right|}\text{ where }A>0. $ Determine its characteristic function.

b) Determine a bound on the probability that a RV is within two standard deviations of its mean.

Solution 1 (retrived from here)

(a)'

$ \Phi_{\mathbf{X}}\left(\omega\right)=E\left[e^{i\omega\mathbf{X}}\right]=\int_{-\infty}^{\infty}\frac{A}{2}e^{-A\left|x\right|}\cdot e^{i\omega x}dx=\frac{A}{2}\left[\int_{-\infty}^{0}e^{x\left(A+i\omega\right)}dx+\int_{0}^{\infty}e^{x\left(-A+i\omega\right)}dx\right] $$ =\frac{A}{2}\left[\frac{e^{x\left(A+i\omega\right)}}{A+i\omega}\biggl|_{-\infty}^{0}+\frac{e^{x\left(-A+i\omega\right)}}{-A+i\omega}\biggl|_{0}^{\infty}\right]=\frac{A}{2}\left[\frac{1}{A+i\omega}-\frac{1}{-A+i\omega}\right] $$ =\frac{A}{2}\cdot\frac{A-i\omega+A+i\omega}{A^{2}+\omega^{2}}=\frac{A^{2}}{A^{2}+\omega^{2}}. $

(b)

$ P\left(\left\{ \left|\mathbf{X}-\overline{\mathbf{X}}\right|\leq2\sigma\right\} \right)=1-P\left(\left\{ \left|\mathbf{X}-\overline{\mathbf{X}}\right|>2\sigma\right\} \right). $ By Chebyshev Inequality, $ P\left(\left\{ \left|\mathbf{X}-\overline{\mathbf{X}}\right|>2\sigma\right\} \right)\leq\frac{\sigma^{2}}{\left(2\sigma\right)^{2}}=\frac{1}{4} $ .

$ P\left(\left\{ \left|\mathbf{X}-\overline{\mathbf{X}}\right|\leq2\sigma\right\} \right)\geq\frac{3}{4}. $

Solution 2

Write it here.

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