Law of large numbers(大数定理)
the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.
当试验次数足够大时,样本均值趋向于总体均值。
实例
以掷骰子为例，已知总体均值为3.5，随着实验次数的增加，样本均值将发生怎样的变化呢？
#!/usr/bin/env python3
#-*- coding:utf-8 -*-
#############################################
#File Name: lln.py
#Brief: 以掷骰子为例，演示随着试验次数的增加，样本均值趋于总体均值
#Author: frank
#Email: [email protected]
#Created Time:2018-08-07 22:29:00
#Blog: http://www.cnblogs.com/black-mamba
#Github: https://github.com/xiaomagejunfu0903/statistic_notes
#############################################
import numpy as np
import matplotlib.pyplot as plt
#总实验次数
total_num = 1000
#骰子的6面
num_sides = 6
plt.figure(1)
#设置x轴的范围
xmin, xmax = plt.xlim(0, total_num)
print("xmin:{},xmax:{}".format(xmin,xmax))
#设置y轴的范围
ymin, ymax = plt.ylim(1, 6)
print("ymin:{},ymax:{}".format(ymin,ymax))
#设置坐标轴的名称
plt.xlabel('number of trials')
plt.ylabel('average')
#画均值线
plt.plot((0,total_num),(3.5,3.5),c='b')
l = []
n_s = range(1, total_num)
#number of trials:n,mean of sample:average
for n in n_s:
average = np.mean(np.random.randint(1,7,n))
l.append(average)
#print("n:{},average:{}".format(n, average))
#plt.scatter(n,average,c='r',s=1,alpha=0.5)
plt.plot(n_s,l,alpha=0.5,c='red')
plt.figure(2)
#设置x轴的范围
ymin, ymax = plt.ylim(0, total_num)
print("ymin:{},ymax:{}".format(ymin,ymax))
#设置y轴的范围
xmin, xmax = plt.xlim(1, 6)
print("xmin:{},xmax:{}".format(xmin,xmax))
#设置坐标轴的名称
plt.xlabel('number of trials')
plt.ylabel('average')
plt.plot((3.5,3.5),(0,total_num),c='b')
#plt.plot(l,n_s,alpha=0.5,c='red')
for n in n_s:
average = np.mean(np.random.randint(1,7,n))
l.append(average)
#print("n:{},average:{}".format(n, average))
plt.scatter(average,n,c='r',s=1,alpha=0.5)
plt.show()
image
image