exercise:9f01957532: 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> Suppose you are watching a radioactive source that emits particles at a rate described by the exponential density <math display="block"> f(t) = \lambda e^{-\lambda t}\ , </math> where <math>\lambda = 1</math>, so that the probability <math>P(0,T)...") |
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\newcommand{\secstoprocess}{\all} | \newcommand{\secstoprocess}{\all} | ||
\newcommand{\NA}{{\rm NA}} | \newcommand{\NA}{{\rm NA}} | ||
\newcommand{\mathds}{\mathbb}</math></div> Suppose you are watching a radioactive source | \newcommand{\mathds}{\mathbb}</math></div> Suppose you are watching a radioactive source that emits particles at a rate described by the exponential density | ||
that emits particles at a rate described by the exponential density | |||
<math display="block"> | <math display="block"> | ||
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in the next <math>T</math> seconds is <math>P([0,T]) = \int_0^T\lambda e^{-\lambda t}\,dt</math>. Find the | in the next <math>T</math> seconds is <math>P([0,T]) = \int_0^T\lambda e^{-\lambda t}\,dt</math>. Find the | ||
probability that a particle (not necessarily the first) will appear | probability that a particle (not necessarily the first) will appear | ||
<ul><li> within the next second. | <ul style="list-style-type:lower-alpha"><li> within the next second. | ||
</li> | </li> | ||
<li> within the next 3 seconds. | <li> within the next 3 seconds. | ||
Latest revision as of 22:36, 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]
Suppose you are watching a radioactive source that emits particles at a rate described by the exponential density
[[math]]
f(t) = \lambda e^{-\lambda t}\ ,
[[/math]]
where [math]\lambda = 1[/math], so that the probability [math]P(0,T)[/math] that a particle will appear in the next [math]T[/math] seconds is [math]P([0,T]) = \int_0^T\lambda e^{-\lambda t}\,dt[/math]. Find the probability that a particle (not necessarily the first) will appear
- within the next second.
- within the next 3 seconds.
- between 3 and 4 seconds from now.
- after 4 seconds from now.