### Poisson Distribution

A discrete random variable X is said to have Poisson distribution if its probability function is defined as,

where Î» is the pararmeter of the distribution and it is the mean number of success. Also Ðµ=2.71828.

Mean of poisson distribution is E[x]= Î» and Variance is Var[X]= Î». The mean and variance of Poisson Distribution is equal.

Poisson curve |

### Properties of Poisson Distribution

Following properties are exist in poission distribution:

- Poisson distribution has only one parameter named "Î»".
- Mean of poisson distribution is Î».
- Poisson is only a distribution which variance is also Î».
- Moment generating function is .
- Poisson distribution is positively skewed and leptokurtic.
- Poisson distribution tends to normal distribution if Î»⟶∞.

### Derivation of Poisson Distribution from Binomial Distribution

Under following condition , we can derive Poission 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⟶∞.
- Î»=np is finite constant that is average number of success is finite.

we can also derive poisson distribution from poisson process.

### Application of poisson distribution

There are some real life example where poisson 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.

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