Basics of Probability Theory - 13 - Poisson Distribution (Poisson Distribution)

This paper records the Poisson distribution.

泊松分布

• 假设已知events in unit time (或者单位面积) 内发生的平均次数为 \lambda, Then the Poisson distribution describes：events in unit time (或者单位面积) The specific number of occurrences within is k 的概率.
• 概率质量函数： p(X=k | \lambda)=\frac{e^{-\lambda} \lambda^{k}}{k !} .
• 期望: \mathbb{E}[X]=\lambda
• 方差: \operatorname{Var}[X]=\lambda

The source of the Poisson distribution

• The number of occurrences per unit time of the Poisson distribution is X,平均次数为\lambda
• Let the observed time period be [0,1),Take a large natural numbern,put the time period[0,1)divided into equal lengthsn段：

l_{1}=\left[0, \frac{1}{n}\right], l_{2}=\left[\frac{1}{n}, \frac{2}{n}\right], \ldots, l_{i}=\left[\frac{i-1}{n}, \frac{i}{n}\right], \ldots, l_{n}=\left[\frac{n-1}{n}, 1\right]

• We make the following two assumptions:

1. 在每段 l_{i} 内, The probability of exactly one accident,Approximate the length of this time \frac{1}{n} 成正比,可设为 \frac{\lambda}{n} .当n很大时, \frac{1}{n} 很小时,在 l_{i} in such a short period of time,It is impossible to have two or more accidents.因此在 l_{i} The probability of no accident during this time period is 1-\frac{\lambda}{n} .
2. l_{i}, \ldots, l_{n} Whether an accident occurs in each segment is independent 把在 [0,1) The number of accidents that occurred during the time period Xregarded as innA small period after the division l_{i}, \ldots, l_{n} The number of time periods in which the accident occurred,According to the above two assumptions, X should be served 从二项分布 B\left(n, \frac{\lambda}{n}\right) .于是,我们有

• 注意到当 n \rightarrow \infty 取极限时,我们有

• 因此

从上述推导可以看出：The Poisson distribution can be obtained as the limit of the binomial distribution.一般的说,若 X \sim B(n, p) ,其中n很大, p很小,因而 n p=\lambda 不太大时, XThe distribution is close to the Poisson distribution P(\lambda) .This fact can sometimes convert the more difficult binomial distribution into a Poisson distribution to calculate.

Python 实现

• scipy The package supports modeling Poisson distributions

查表

Check the cumulative probability.查询 \lambda =100,The number of occurrences is less than or equal to120的概率：

from scipy import stats
 
p = stats.poisson.cdf(120, 100)
print(p)

>>>
0.9773306709216473

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随机数生成

生成服从

=50的泊松分布随机数100个：

from scipy import stats
# 设置random_state时,The random number generated each time is the same.不设置或为None时,The random numbers generated multiple times are different
sample = stats.poisson.rvs(mu=50, size=100, random_state=3)
print(sample)

>>>
[51 45 60 40 34 53 54 45 45 49 51 46 48 61 47 53 47 48 45 49 52 45 43 50
 50 54 54 47 47 46 36 72 54 55 52 37 42 41 54 54 55 58 53 53 51 43 58 38
 63 50 44 53 48 43 53 45 67 37 51 42 54 47 59 55 54 55 55 46 60 43 54 45
 59 44 58 45 51 58 56 47 54 33 55 50 58 49 60 37 51 43 50 52 52 45 42 44
 49 54 52 48]

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参考资料

• http://www.huaxiaozhuan.com/数学基础/chapters/2_probability.html
• https://baike.baidu.com/item/泊松分布/1442110?fr=aladdin
• https://blog.csdn.net/lanhezhong/article/details/105765526