二项式和泊松分布

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最新推荐文章于 2024-06-29 15:06:32 发布

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原文链接：https://medium.com/@akhilanandkspa/binomial-and-poisson-distribution-337958a384f3

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二项分布是一种离散分布，用于描述成功或失败的事件，如考试的通过/失败或硬币的正反面。它与正态分布不同，但在大量数据时，二项分布趋向于正态分布的行为。本文将探讨二项分布的图形表示及与正态分布的比较，并介绍其数学公式和参数。

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二项分布(Binomial Distribution)

What is Binomial Distribution ?

什么是二项分布？

It is a discrete distribution and describes success or failure of an event. e.g:- In an examination student can either pass or fail , if a coin is tossed it gives either head or tail. In other word Binomial Distribution deals with only two possible outcomes.

它是离散分布，描述事件的成功或失败。例如：-在考试中，学生可以通过或不通过，如果掷硬币，则正面或反面都可以。换句话说，二项分布只处理两个可能的结果。

Difference between Binomial Distribution and normal Distribution ?

二项分布与正态分布之间的区别？

Binomial Distribution is Discrete whereas Normal Distribution is continious in nature but for a large datapoints Binomial Distribution predominantly behaves like Normal Distribution.

二项分布是离散的，而正态分布在本质上是连续的，但对于较大的数据点，二项分布的行为主要类似于正态分布。

pictorial representation

图形表示

import numpy as np   #Binomial Distribution plot
import matplotlib.pyplot as plt
import seaborn as sns
binomial_data=np.random.binomial(n=10,p=0.3,size=1000)
sns.distplot(binomial_data,kde=True,hist=True,color="green")

comparison between binomial and normal distriburtion

二项分布与正态分布的比较

import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
sns.distplot(np.random.binomial(n=100,p=0.3,size=100),hist=False,kde=True,color="green",label="Binomial")
sns.distplot(np.random.normal(loc=30,scale=5,size=100),hist=False,kde=True,color="red",label="Normal")

Mathematical formulation and Parameters of Binomial Distribution(n,p,size,x)

二项分布(n，p，size，x)的数学公式和参数

Parameters P,Q,n,x can be defined in next subsection with the help of an example.

参数P，Q，n，x可以在下一节中借助示例进行定义。

Q 1.> A company manufactures LED bulbs with a faulty rate of 30%. If I randomly select 6 chosen LEDs, what is the probability of having 2 faulty LEDs in my sample? Calculate the average value of this process. Also evaluate the standard deviation associated with it ?

问1.>一家公司生产的LED灯泡的故障率为30％。如果我随机选择6个LED，样本中有2个故障LED的概率是多少？计算此过程的平均值。还评估与之相关的标准偏差吗？

Defining parameters;

定义参数；

P=0.3     ;   Q=1-P=0.7
n=total number of trials=6
k=number of trail that will be successed=2
size=Total number of random samples =1000

Mathematical Calculation :-

数学计算：

why we take random samples(z) ?

为什么我们要抽取随机样本(z)？

When we analyse data as a M.L engineer we must ensure how the uncertanity introduced by random samples affects our datasets .We also try to evaluate how data would be affected by random error.

当我们以ML工程师的身份分析数据时，必须确保随机样本引入的不可靠性如何影响我们的数据集。我们还尝试评估随机误差将如何影响数据。

Python implementation and plotting

Python实现和绘图

from scipy.stats import binom
import matplotlib.pyplot as plt
import seaborn as sns
binomial_data=binom.rvs(n=6,p=0.3,size=1000)
sns.distplot(binomial_data,hist=True,kde=True,color="red")

#Probability of getting faulty out of 6 trials
from scipy.stats import binom
import matplotlib.pyplot as plt
import seaborn as sns
probab=binom.pmf(k=2,n=6,p=0.3)
print("Probability will be :",probab)
cdf=binom.cdf(k=2,n=6,p=0.3)
print("CDF will be :",cdf)[out]>> Probability will be : 0.32413499999999995
        CDF will be : 0.74431

Now I will calculate mean and standard deviation;

现在，我将计算均值和标准差；

mean=np & standard deviation=sqrt(npq)

平均值= np和标准偏差= sqrt(npq)

import scipy.stats as binom
import math 
mean,var=binon.stats(n=6,p=0.3)
print("mean := ',mean)
print("standard deviation :=",math.sqrt(var))[out]>>mean :=  1.7999999999999998
standard deviation := 1.1224972160321822

泊松分布 (Poisson Distribution)

What is poisson distribution ?

什么是泊松分布？

It describe the distribution of rare event in a given population.It is mainly used for forecasting eg:- how many pligrims have been visited to vaishno devi during covid-19 pandemic etc.

它描述了给定人群中稀有事件的分布，主要用于预测例如：-在covid-19大流行等期间曾拜访过vaishno devi的朝圣者。

How to decide when to use binomial or Poisson ?

如何决定何时使用二项式或泊松？

i. > Suppose you have given average number of probability per unit time and you have to find certain number of probability for a particular time then poisson is used.

一世。 >假设您给出了每单位时间的平均概率数，并且必须找到特定时间的一定概率数，然后使用泊松法。

ii.> If you have given exact probability and need to find out probability happening certain number of time out of 10 times, 100 times etc.

ii。>如果您已经给出了确切的概率，并且需要找出概率发生在10次，100次等中的一定次数，

Mathematical formulation and parametric understanding

数学公式化和参数理解

lambda:- mean number of occurances in the interval.

lambda：-间隔中的平均发生次数。

x/k:-number of success we are interested in

x / k：-我们感兴趣的成功次数

Q..>Customers arrive at a rate of 72 per hour to my shop. What is the probability of 𝑘 customers arriving in 4 minutes? a) 5 customers, b) not more than 3 customers.

问：>顾客到我店的速度是每小时72个。 𝑘客户在4分钟内到达的概率是多少？ a)5个客户，b)不超过3个客户。

sol:- customer arrive per minute=72/60=1.2 then, in 4 min total number of customer will arrive =1.2 X 4=4.8

sol：-客户每分钟到达= 72/60 = 1.2，那么，在4分钟内客户总数将到达= 1.2 X 4 = 4.8

now applying poisson formula we will get;

现在应用泊松公式，我们将得到：

pictorial presentation using python

使用python的图片演示

from scipy.stats import poisson
import matplotlib.pyplot as plt
import seaborn as sns
poisson_data=poisson.rvs(mu=4.8,size=1000)
sns.distplot(poisson_data,hist=True,kde=True,color="red")

from scipy.stats import poisson
probab1=poisson.pmf(k=5,mu=4.8)
probab2=sum(poisson.pmf(k=[0,1,2],mu=4.8))
print(probab1,probab2)[out]>> 0.17474768364388296 0.14253921888902693