exercise:444d73c222: 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> Let <math>X</math> be a random variable distributed uniformly over <math>[0,20]</math>. Define a new random variable <math>Y</math> by <math>Y = \lfloor X\rfloor</math> (the greatest integer in <math>X</math>). Find the expected value of <math>Y...") |
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Let <math>X</math> be a random variable distributed uniformly over <math>[0,20]</math>. Define a new random variable <math>Y</math> by <math>Y = \lfloor X\rfloor</math> (the greatest | |||
<math>[0,20]</math>. Define a new random variable <math>Y</math> by <math>Y = \lfloor X\rfloor</math> (the greatest | |||
integer in <math>X</math>). Find the expected value of <math>Y</math>. Do the same for <math>Z = | integer in <math>X</math>). Find the expected value of <math>Y</math>. Do the same for <math>Z = | ||
\lfloor X + .5\rfloor</math>. Compute <math>E\bigl(|X-Y|\bigr)</math> and <math>E\bigl(|X-Z|\bigr)</math>. | \lfloor X + .5\rfloor</math>. Compute <math>E\bigl(|X-Y|\bigr)</math> and <math>E\bigl(|X-Z|\bigr)</math>. | ||
Latest revision as of 22:38, 14 June 2024
Let [math]X[/math] be a random variable distributed uniformly over [math][0,20][/math]. Define a new random variable [math]Y[/math] by [math]Y = \lfloor X\rfloor[/math] (the greatest integer in [math]X[/math]). Find the expected value of [math]Y[/math]. Do the same for [math]Z = \lfloor X + .5\rfloor[/math]. Compute [math]E\bigl(|X-Y|\bigr)[/math] and [math]E\bigl(|X-Z|\bigr)[/math]. (Note that [math]Y[/math] is the value of [math]X[/math] rounded off to the nearest smallest integer, while [math]Z[/math] is the value of [math]X[/math] rounded off to the nearest integer. Which method of rounding off is better? Why?)