Poisson-Distribution

Poisson Distribution

  • Consider the number of times an event occurs over an interval of time or space, and assume that: (考虑事件在一段时间和空间内发生的次数)

      1. The probability of occurrence is the same for any intervals of equal length
      1. The occurrence in any interval is independent of an occurrence in any non-overlapping interval

  • The probability distribution of X is characterized by

  • For Poisson distribution, the following is true:
    λ is called the Poisson parameter. λ is the average number of items considered in the Poisson process during the observed time period.

From Binomial distribution to Poisson distribution

  • Binomial distribution
  • Poisson distribution

Intuitively, the Binomial distribution and the Poisson distribution seem to be unrelated. But a closer look reveals a very interesting relationship. The Poisson distribution is just a special case of the binomial distribution. When n tends to infinity and p approaches zero, the binomial distribution approaches the Poisson distribution.

As you can see on the slide,  intuitively the 2 distributions seem to be unrelated. However, by searching on the internet. We found an interesting relationship between the Binomial distribution and the Poisson distribution. That is when the trail n tends to infinity.  The Poisson distribution is just a special case of the binomial distribution.

Derivation

Let me give you a simple derivation

For binomial distribution


    • represents the number of trails
    • represents the probability that a trial will succeed
As you can see the binomial distribution on the slide, the expected value of the binomial distribution is equal to the variable n times the probability p. So we can get p by lambda divided by n.

Put the equation in the expression for the binomial distribution， we can get the following expression

And we put this equation to the binomial distribution, we can get the following expression

By expanding the combination number

There is an assumption in the Poisson distribution

For Poisson distribution 

For the Poisson distribution, the variable n tends to the infinity
So we just need to simplify this limit expression

We can expand the factorial， then cancel out the (n-k)factorial part

After that, we switch the positions of the denominators

We can see that the factorial part has k terms， and we have n k times

This part can be divided into two terms

According to the definition of the natural constant e

It can be found that the base part is the same as the definition

Table Of Contents
Poisson-Distribution
Poisson Distribution
From Binomial distribution to Poisson distribution
Derivation