exercise:88d9acf85f: Difference between revisions
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(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...") |
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A number is chosen at random from the integers <math>1, 2, 3,\ldots, n</math>. Let | |||
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<math>X</math> be the number chosen. Show that <math>E(X) = (n + 1)/2</math> and <math>V(X) = (n - 1)(n + | <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>. | 1)/12</math>. '' Hint'': The following identity may be useful: | ||
<math display="block"> | <math display="block"> | ||
1^2 + 2^2 + \cdots + n^2 = \frac{(n)(n+1)(2n+1)}{6}\ . | 1^2 + 2^2 + \cdots + n^2 = \frac{(n)(n+1)(2n+1)}{6}\ . | ||
</math> | </math> | ||
Latest revision as of 22:03, 14 June 2024
A number is chosen at random from the integers [math]1, 2, 3,\ldots, 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]]