exercise:Dc42ff4e62: Difference between revisions
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(Created page with "<div class="d-none"><math> \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}</math></div> Let <math>X</math> be a continuous random variable whose characteristic function <math>k_X(\tau)</math> is <math display="block"> k_X(\tau) = e^{-|\tau|}, \qquad -\infty < \tau < +\infty\ . </math> Show directly that the density <math>f_X</ma...") |
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Let <math>X</math> be a continuous random variable whose characteristic function | |||
<math>k_X(\tau)</math> is | <math>k_X(\tau)</math> is | ||
Latest revision as of 01:05, 15 June 2024
Let [math]X[/math] be a continuous random variable whose characteristic function [math]k_X(\tau)[/math] is
[[math]]
k_X(\tau) = e^{-|\tau|}, \qquad -\infty \lt \tau \lt +\infty\ .
[[/math]]
Show directly that the density [math]f_X[/math] of [math]X[/math] is
[[math]]
f_X(x) = \frac1{\pi(1 + x^2)}\ .
[[/math]]