Physics307L F08:Schedule/Week 12 agenda/PDFs

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Lots of algorithms available for generating random numbers from 0 to 1 with uniform probability

General case

Assume you have a random number r from 0 to 1. To transform to x, with PDF given by P(x):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle r=\int_{X=-\infty}^{x} P(x)\, dx}

Numerically solving for x gives random number with appropriate distribution.

Specific case, time for random event to occur (i.e. radioactive decay)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle P(t) = \frac {1}{\tau} e^{-\frac{t}{\tau}}}

Solving above,

<math>\ t_i = -\tau \ln r_i