exercise:A9f9e8427b: Difference between revisions

Latest revision as of 21:36, 14 June 2024

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].

@@ Line 1: / Line 1: @@
-<div class="d-none"><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|  <  1</math>.
-\newcommand{\NA}{{\rm NA}}
+<ul style="list-style-type:lower-alpha"><li> Show that if <math>\int_{-1}^{+1} f_X(x)\, dx = 1</math>, then <math>b = 1/2</math>.
-\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>
 <li> Show that if <math>f_X(x) \geq 0</math>, then <math>-1/2 \leq a \leq 1/2</math>.