Physics307L F09:Schedule/Week 10 agenda/Poisson

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Poisson Distribution

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(k;\lambda)=\frac{\lambda^k e^{-\lambda}}{k!},\,\!} (This is a probability mass function)

Is the limit of the bionomial distribution when probability of success goes to zero, number of trials goes to infinity, and p*n = lambda

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 \ \sigma_{k}\, =\, \sqrt{\lambda}}

For a given collection of data, thought to be Poisson distributed, the maximum likelihood fit is

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 \lambda = \frac {\sum{x_i}}{N}, }

where x_i are the counts recorded in each trial, and N is the number of trials

Example: decay of radioactive sample

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_\mathrm{Poisson}(k;\lambda) \approx p_\mathrm{normal}(k;\mu=\lambda,\sigma^2=\lambda)\,}