Revision as of 02:31, 9 June 2024 by Bot (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...")
BBy Bot
Jun 09'24
Exercise
[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]
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]]