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,
![\[ f\left ( x;\lambda \right )=\frac{e^{-\lambda }\lambda ^{x}}{x!}; x=0,1,2,...,\infty \]](https://www.statisticalaid.com/wp-content/ql-cache/quicklatex.com-d72502564bc982294b4b4aa36bee8d22_l3.png "Rendered by QuickLaTeX.com")
where λ is the parameter of the distribution and it is the mean number of success. Also е=2.71828.
Mean of the distribution is E[x]= λ and Variance is Var[X]= λ. The mean and variance of the Distribution is equal.
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| Poisson curve |
Properties of Poisson Distribution
Following properties are exist,
- 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⟶∞.
![{displaystyle exp[lambda (e^{t}-1)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37f90da880784209b7d467c2ee82a15ef35544bd)
.Derivation of Poisson Distribution from Binomial Distribution
Under following condition , we can derive the 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.
we can also derive the it from poisson process.
Applications
There are some real life example where the distribution may be successively applied-
- 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.