exercise:B8ae29be7e: 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> Assume that a new light bulb will burn out after <math>t</math> hours, where <math>t</math> is chosen from <math>[0,\infty)</math> with an exponential density <math display="block"> f(t) = \lambda e^{-\lambda t}\ . </math> In this context, <math>...") |
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\newcommand{\mathds}{\mathbb}</math></div> Assume that a new light bulb will burn out after <math>t</math> | \newcommand{\mathds}{\mathbb}</math></div> Assume that a new light bulb will burn out after <math>t</math> hours, where <math>t</math> is chosen from <math>[0,\infty)</math> with an exponential density | ||
hours, where <math>t</math> is chosen from <math>[0,\infty)</math> with an exponential density | |||
<math display="block"> | <math display="block"> | ||
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</math> | </math> | ||
In this context, <math>\lambda</math> is often called the ''failure rate'' of the bulb. | In this context, <math>\lambda</math> is often called the ''failure rate'' of the bulb. | ||
<ul><li> Assume that <math>\lambda = 0.01</math>, and find the probability that the bulb | <ul style="list-style-type:lower-alpha"><li> Assume that <math>\lambda = 0.01</math>, and find the probability that the bulb | ||
will ''not'' burn out before <math>T</math> hours. This probability is often called | will ''not'' burn out before <math>T</math> hours. This probability is often called | ||
the ''reliability'' of the bulb. | the ''reliability'' of the bulb. | ||
Latest revision as of 21:37, 12 June 2024
[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]
Assume that a new light bulb will burn out after [math]t[/math] hours, where [math]t[/math] is chosen from [math][0,\infty)[/math] with an exponential density
[[math]]
f(t) = \lambda e^{-\lambda t}\ .
[[/math]]
In this context, [math]\lambda[/math] is often called the failure rate of the bulb.
- Assume that [math]\lambda = 0.01[/math], and find the probability that the bulb will not burn out before [math]T[/math] hours. This probability is often called the reliability of the bulb.
- For what [math]T[/math] is the reliability of the bulb [math] = 1/2[/math]?