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
  • 2. 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.

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

$ C^k_n \ (\frac{\lambda}{n})^k \ (1-(\frac{\lambda}{n}))^{n-k} $

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