Poisson Distribution: Definition, Properties and applications with real life example

Poisson Distribution

Poisson distribution by statisticalaid.com

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

Poission curve
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.

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