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 pararmeter 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.

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⟶∞.

### 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.

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