Physics307L F07:Schedule/Week 12 agenda/Poisson

Poisson Distribution

[math]\displaystyle{ p(k;\lambda)=\frac{\lambda^k e^{-\lambda}}{k!},\,\! }[/math]

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]

Example: decay of radioactive sample

[math]\displaystyle{ p_\mathrm{Poisson}(k;\lambda) \approx p_\mathrm{normal}(k;\mu=\lambda,\sigma^2=\lambda)\, }[/math]