Revision as of 02:24, 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> A number is chosen at random from the integers 1, 2, 3, \dots, <math>n</math>. Let <math>X</math> be the number chosen. Show that <math>E(X) = (n + 1)/2</math> and <math>V(X) = (n - 1)(n + 1)/12</math>. '' Hint'': The following identity may b...")
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]
A number is chosen at random from the integers 1, 2, 3,
\dots, [math]n[/math]. Let [math]X[/math] be the number chosen. Show that [math]E(X) = (n + 1)/2[/math] and [math]V(X) = (n - 1)(n + 1)/12[/math]. Hint: The following identity may be useful:
[[math]]
1^2 + 2^2 + \cdots + n^2 = \frac{(n)(n+1)(2n+1)}{6}\ .
[[/math]]