Create Time: 30th Dec 2023
Title: "[[Poisson-Distribution]]"
status: DONE
Author:
- AllenYGY
tags:
- Poisson
- Distribution
- ProbabilityStatics
- NOTE
created: 2024-01-16T21:03
updated: 2024-04-08T19:58
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)
The probability distribution of X is characterized by
For Poisson distribution, the following is true:
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.
Let me give you a simple derivation
For binomial distribution
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
$ 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
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