Physics307L:Schedule/Week 10 agenda/Poisson
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Poisson Distribution
- [math]\displaystyle{ p(k;\lambda)=\frac{\lambda^k e^{-\lambda}}{k!},\,\! }[/math] (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
[math]\displaystyle{ \ \sigma_{k}\, =\, \sqrt{\lambda} }[/math]
For a given collection of data, thought to be Poisson distributed, the maximum likelihood fit is
- [math]\displaystyle{ \lambda = \frac {\sum{x_i}}{N}, }[/math]
where x_i are the counts recorded in each trial, and N is the number of trials
Example: decay of radioactive sample
- [math]\displaystyle{ p_\mathrm{Poisson}(k;\lambda) \approx p_\mathrm{normal}(k;\mu=\lambda,\sigma^2=\lambda)\, }[/math]