The Poisson distribution is the discrete probability distribution of the number of events occurring in a given time period, given the average number of times the event occurs over that time period.
A discrete random variable X is said to have Poisson distribution if its probability function is defined as,
where λ is the parameter of the distribution and it is the mean number of success. Also е=2.71828.
Properties of Poisson Distribution
- Poisson distribution has only one parameter named “λ”.
- Mean of poisson distribution is λ.
- It is only a distribution which variance is also λ.
- Moment generating function is .
- The distribution is positively skewed and leptokurtic.
- It tends to normal distribution if λ⟶∞.
Derivation of Poisson Distribution from Binomial Distribution
- The probability of success or failure in bernoulli trial is very small that means which tends to zero. p⟶0 and q⟶0.
- The number of trial n is very large that is n⟶∞.
- λ=np is finite constant that is average number of success is finite.
- Number of death from a disease such as cancer or heart attack.
- Number of suicide reports in a perticular day.
- Number of printing mistake at each page of a book.
- Number of car passing through a certain road in a period of time t.
- Number of road accident in a certain street at a time t etc.