# Physics307L:Schedule/Week 10 agenda/Poisson

## Poisson Distribution

$p(k;\lambda)=\frac{\lambda^k e^{-\lambda}}{k!},\,\!$ (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

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

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

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

### Example: decay of radioactive sample

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