# Summary of mathematical statistical confidence intervals

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A single normal population

Two normal populations

three   Interval estimation of (0-1) parameters

one   A normal population

The values here are the upper alpah quantile

 Parameters to be estimated Other parameters Distribution of pivot amount Bilateral confidence interval Unilateral upper limit Unilateral lower limit u Known u unknown u unknown

Example:   Five bulbs are randomly selected from a batch of bulbs for life test, and the life is measured

1050 1100 1120 1250 1280

Let the bulb obey the normal distribution, and find the lower limit of one-sided confidence interval with the average confidence level of 0.95

```# -*- coding: utf-8 -*-
"""
Created on Sat Nov 20 20:54:34 2021

@author: cxf
"""
import numpy as np
from scipy.stats import t
import matplotlib.pyplot as plt

class Confidence():

def GetData(self):

data =[1050, 1100,1120,1250,1280]

x = np.arange(1,len(data)+1)

plt.bar(x, data,width=0.2, label='use year')
plt.xlabel("item ")
plt.ylabel("year")
plt.legend()
plt.show()
return data

def __init__(self):
self.alpha = 0.05 #The confidence interval was 0.95

'''
ddof= 1 Is the sample standard deviation, otherwise it is the overall standard deviation
'''
def GetConfidence(self):

data = self.GetData()
n = len(data)
x_bar = np.mean(data)
s = np.std(data,ddof=1)

v = t.isf(self.alpha,n-1)
print("\n Sample mean %5.2f  ,Sample standard deviation %5.2f  Number of samples %d , upper alpha Quantile%5.2f "%(x_bar, s,n,v))
low = x_bar-s/np.sqrt(n)*v

print("\n Lower limit of unilateral confidence interval u %5.2f"%low)

if __name__ == "__main__":

Co = Confidence()

Co.GetConfidence()
=======================================

Sample mean 1160.00  ,Sample standard deviation 99.75  Number of samples 5 , upper alpha Quantile 2.13

Lower limit of unilateral confidence interval u 1064.90```

Two normal populations

 Parameters to be estimated Other parameters Distribution of pivot amount Bilateral confidence interval Unilateral upper limit Unilateral lower limit u1-u2 Known u1-u2 unknown unknown

three   Interval estimation of (0-1) distribution parameters

Sample mean u, variance p(1-p)

Distribution of pivot amount:

confidence interval

Binomial distribution is

In probability theory and statistics, binomial distribution is a discrete probability distribution of the number of successes in n independent success / failure tests, in which the success probability of each test is p. Such a single success / failure test is also called Bernoulli test . In fact, when n=1, the binomial distribution is Bernoulli distribution.

For example, in the KNN algorithm, the general algorithm does not stipulate that we finally take several K numbers from N numbers.

Because the number of K is closest to the Euclidean distance of the sample, D(Xk)=0 is expected to be a type.

For example, if the label type of the sample is 5, the probability of selecting a random sample in k and the label value p of the data itself is

1/N.

As can be seen from Fig. 1, for fixed n and P, when k increases, the probability P{X=k} first increases until it reaches the maximum, and then decreases monotonically. It can be proved that the general binomial distribution also has this property, and:   [1]

1. When (n+1) P is not an integer, the binomial probability P{X=k} reaches the maximum when k=[(n+1)p];   [1]

2. When (n+1) P is an integer, the binomial probability P{X=k} reaches the maximum when k=(n+1)p and k=(n+1)p-1.

example

```# -*- coding: utf-8 -*-
"""
Created on Sun Nov 21 16:08:06 2021

@author: cxf
"""

import matplotlib.pyplot  as plt
import numpy as np
import scipy.misc as ms
from scipy.special  import comb ,perm

def GetP(p,m,n,cb):

a = cb*np.power(p,n)*np.power(1-p,m-n)
return a

def GetData(m,p):
y =[]
x = np.arange(0,m)

for  n in range(m):
a = comb(m,n) #Permutation and combination
b = np.power(p,n)
c = np.power(1.0-p,m-n)
#print("cb ",False)
d = a*b*c
y.append(d)

print("prob ",p)
plt.scatter(x,y,c='r')
plt.show()

def DrawTwo():
a= 0

GetData(100,0.2)

DrawTwo()
```

Posted by TonyB on Sun, 21 Nov 2021 18:37:18 -0800