# Poisson Distribution: Definition, Properties and application with real life example

### Poisson Distribution

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

$f\left&space;(&space;x;\lambda&space;\right&space;)=\frac{e^{-\lambda&space;}\lambda&space;^{x}}{x!};&space;where&space;;x=0,1,2...,\infty$

where λ is the pararmeter of the distribution and it is the mean number of success. Also е=2.71828.

Mean of poisson distribution is E[x]= λ and Variance is Var[X]= λ. 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 "λ".
• Mean of poisson distribution is λ.
• Poisson is only a distribution which variance is also λ.
• Moment generating function is .
• Poisson distribution is positively skewed and leptokurtic.
• Poisson distribution tends to normal distribution if λ⟶∞.

### 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⟶∞.
• λ=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.