Revision as of 02:25, 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 random variable with range <math>[-1,1]</math> and density function <math>f_X(x) = ax + b</math> if <math>|x| < 1</math>. <ul><li> Show that if <math>\int_{-1}^{+1} f_X(x)\, dx = 1</math>, then <math>b = 1/2</math>. </li>...")
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 random variable with range [math][-1,1][/math] and density
function [math]f_X(x) = ax + b[/math] if [math]|x| \lt 1[/math].
- Show that if [math]\int_{-1}^{+1} f_X(x)\, dx = 1[/math], then [math]b = 1/2[/math].
- Show that if [math]f_X(x) \geq 0[/math], then [math]-1/2 \leq a \leq 1/2[/math].
- Show that [math]\mu = (2/3)a[/math], and hence that [math]-1/3 \leq \mu \leq 1/3[/math].
- Show that [math]\sigma^2(X) = (2/3)b - (4/9)a^2 = 1/3 - (4/9)a^2[/math].