exercise:Fa74d77804: 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> If <math>X</math> is a random variable with mean <math>\mu \ne 0</math> and variance <math>\sigma^2</math>, define the ''relative deviation'' <math>D</math> of <math>X</math> from its mean by <math display="block"> D = \left| \frac {X - \mu}\mu \...") |
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If <math>X</math> is a random variable with mean <math>\mu \ne 0</math> and variance <math>\sigma^2</math>, define the ''relative deviation'' <math>D</math> of <math>X</math> from its mean by | |||
define the ''relative deviation'' <math>D</math> of <math>X</math> from its mean by | |||
<math display="block"> | <math display="block"> | ||
D = \left| \frac {X - \mu}\mu \right|\ . | D = \left| \frac {X - \mu}\mu \right|\ . | ||
</math> | </math> | ||
<ul><li> Show that <math>P(D \geq a) \leq \sigma^2/(\mu^2a^2)</math>. | <ul style="list-style-type:lower-alpha"><li> Show that <math>P(D \geq a) \leq \sigma^2/(\mu^2a^2)</math>. | ||
</li> | </li> | ||
<li> If <math>X</math> is the random variable of Exercise [[exercise:63ec956b0e |Exercise]], find an | <li> If <math>X</math> is the random variable of Exercise [[exercise:63ec956b0e |Exercise]], find an | ||
Latest revision as of 22:48, 14 June 2024
If [math]X[/math] is a random variable with mean [math]\mu \ne 0[/math] and variance [math]\sigma^2[/math], define the relative deviation [math]D[/math] of [math]X[/math] from its mean by
[[math]]
D = \left| \frac {X - \mu}\mu \right|\ .
[[/math]]
- Show that [math]P(D \geq a) \leq \sigma^2/(\mu^2a^2)[/math].
- If [math]X[/math] is the random variable of Exercise Exercise, find an upper bound for [math]P(D \geq .2)[/math], [math]P(D \geq .5)[/math], [math]P(D \geq .9)[/math], and [math]P(D \geq 2)[/math].