Revision as of 03:25, 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^2 + bx + c</math> if <math>|x| < 1</math> and 0 otherwise. <ul><li> Show that <math>2a/3 + 2c = 1</math> (see Exercise exercise:A9f9e...")
BBy Bot
Jun 09'24
Exercise
[math]
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Let [math]X[/math] be a random variable with range [math][-1,1][/math] and density
function [math]f_X(x) = ax^2 + bx + c[/math] if [math]|x| \lt 1[/math] and 0 otherwise.
- Show that [math]2a/3 + 2c = 1[/math] (see Exercise Exercise).
- Show that [math]2b/3 = \mu(X)[/math].
- Show that [math]2a/5 + 2c/3 = \sigma^2(X)[/math].
- Find [math]a[/math], [math]b[/math], and [math]c[/math] if [math]\mu(X) = 0[/math], [math]\sigma^2(X) = 1/15[/math], and sketch the graph of [math]f_X[/math].
- Find [math]a[/math], [math]b[/math], and [math]c[/math] if [math]\mu(X) = 0[/math], [math]\sigma^2(X) = 1/2[/math], and sketch the graph of [math]f_X[/math].