第3章のPythonコード

第3章 確率論の基礎

サンプルデータ

wage.csv：男性労働者の賃金データ．

モジュールのインポート

import numpy as np
import math
import matplotlib.pyplot as plt
import pandas as pd
import scipy.stats as stats
import seaborn as sns

乱数の設定

np.random.seed(seed = 2022)

3.1.2 試行と事象

coin = ['Head', 'Tail']
np.random.choice(coin, size = 100, replace = True)
array(['Tail', 'Head', 'Tail', 'Head', 'Tail', 'Tail', 'Head', 'Tail',
       'Head', 'Head', 'Head', 'Head', 'Tail', 'Tail', 'Tail', 'Tail',
       'Tail', 'Tail', 'Head', 'Head', 'Head', 'Tail', 'Head', 'Tail',
       'Head', 'Head', 'Head', 'Tail', 'Head', 'Head', 'Head', 'Tail',
       'Head', 'Tail', 'Head', 'Head', 'Tail', 'Tail', 'Tail', 'Tail',
       'Tail', 'Tail', 'Tail', 'Tail', 'Head', 'Head', 'Head', 'Head',
       'Head', 'Tail', 'Head', 'Tail', 'Tail', 'Head', 'Head', 'Head',
       'Head', 'Tail', 'Tail', 'Head', 'Head', 'Tail', 'Head', 'Tail',
       'Tail', 'Tail', 'Tail', 'Tail', 'Tail', 'Tail', 'Head', 'Tail',
       'Tail', 'Tail', 'Head', 'Head', 'Head', 'Head', 'Tail', 'Head',
       'Head', 'Tail', 'Head', 'Tail', 'Tail', 'Head', 'Head', 'Tail',
       'Tail', 'Tail', 'Tail', 'Head', 'Tail', 'Tail', 'Tail', 'Tail',
       'Head', 'Tail', 'Tail', 'Head'], dtype='<U4')

3.1.3 コイン投げのシミュレーション

math.factorial(100) / (math.factorial(50) ** 2) / (2 ** 100)
0.07958923738717877
coin = [1, 0]
z = np.random.choice(coin, size = 100, replace = True)

z.sum()
53
S = 100000
rec = np.zeros(S)
coin = [1, 0]

for i in range(S):
  z = np.random.choice(coin, 100, replace = True)
  rec[i] = z.sum()
plt.hist(rec, bins = 20)
plt.show()

pd.DataFrame(rec).describe()
0
count 100000.000000
mean 49.986290
std 4.987395
min 28.000000
25% 47.000000
50% 50.000000
75% 53.000000
max 74.000000

3.1.4 論理演算によるカウントの方法

2 > 1
True
2 > 1000
False
200 == 100 * 2
True
True + True
2
True * False
0
count = (rec == 50)
print(count)
[False False False ... False False False]
count.sum()
7997
count.mean()
0.07997

3.2.3 独立ではない例

# 独立性の確認

S = 10000 # シミュレーション回数
X = np.random.normal(loc = 50, scale = 10, size = S) # Xを抽出
Y = np.random.normal(loc = 50, scale = 10, size = S) # Yを抽出
Z = X + Y # Zを構成
# Pr(X > 70) * Pr(Z > 100)
(X > 70).mean() * (Z > 100).mean() 
0.010649170000000001
# Pr(X > 70 かつ Z > 100)
((X > 70) * (Z > 100)).mean() 
0.0207

3.2.4 独立性と相関係数

X = np.random.normal(loc = 50, scale = 10, size = 100000)
Y = np.random.normal(loc = 50, scale = 10, size = 100000)

np.corrcoef(X, Y)
array([[ 1.       , -0.0090849],
       [-0.0090849,  1.       ]])
Z = -((X - 50) ** 2) / 10
np.corrcoef(X, Z)
array([[1.        , 0.00952884],
       [0.00952884, 1.        ]])
plt.scatter(X, Z)
plt.show()

3.3.1 分布関数

x = np.linspace(0, 100, 1000)
plt.plot(x, stats.norm.cdf(x, 50, 10))
plt.show()

print(stats.norm.cdf(60, 50, 10) - stats.norm.cdf(40, 50, 10))
0.6826894921370859

3.3.2 確率密度関数

print(stats.norm.cdf(50 + 0.1, 50, 10) - stats.norm.cdf(50, 50, 10))
0.003989356314631709
print(stats.norm.pdf(50, 50, 10))
0.03989422804014327
print(stats.norm.pdf(80, 50, 10))
0.00044318484119380076
x = np.linspace(start = 0, stop = 100, num = 1000)
plt.plot(x, stats.norm.pdf(x, 50, 10))
plt.show()

3.3.7 データによる条件付期待値の推定

wagedata = pd.read_csv("wage.csv")
wagedata.head()
educ exper wage
0 7 16 548.000002
1 12 9 481.000000
2 12 16 721.000002
3 11 10 250.000001
4 12 16 728.999999 
wagedata.describe()
educ exper wage
count 3010.000000 3010.000000 3010.000000
mean 13.263455 8.856146 577.282392
std 2.676913 4.141672 262.958303
min 1.000000 0.000000 100.000000
25% 12.000000 6.000000 394.250000
50% 13.000000 8.000000 537.499999
75% 16.000000 11.000000 708.750001
max 18.000000 23.000000 2404.000010 
sns.pairplot(wagedata)
<seaborn.axisgrid.PairGrid at 0x11b719360>

educ12 = wagedata.loc[wagedata['educ'] == 12]
educ16 = wagedata.loc[wagedata['educ'] == 16]
educ12['wage'].mean()
563.5342744186569
educ16['wage'].mean()
642.8932464464075
educ11_less = wagedata.loc[wagedata['educ'] < 12]
educ12_more = wagedata.loc[wagedata['educ'] >= 12]
educ11_less['wage'].mean()
438.10865197516847
educ12_more['wage'].mean()
604.8070038470858

3.4.2 中心極限定理のシミュレーション

S = 10000
n = 10000
Zn = np.zeros(S)

for i in range(S):
  X = np.random.normal(loc = 50, scale = 10, size = n)
  Xbar = X.mean()
  Sn = X.var()
  Zn[i] = (n ** 0.5) * (Xbar - 50) / (Sn ** 0.5)

plt.hist(Zn, bins = 20)
plt.show()

pd.DataFrame(Zn).describe()
0
count 10000.000000
mean 0.010709
std 1.000739
min -3.803410
25% -0.663426
50% 0.015367
75% 0.688068
max 3.487146

3.4.3 信頼区間のシミュレーション

# 信頼区間シミュレーション

S = 10000 # シミュレーション回数
n = 10000 # 標本の大きさ
rec = np.zeros(S) # 結果記録用のベクトル

for i in range(S):  # 繰り返し開始
  X = np.random.normal(loc = 50, scale = 10, size  = n)  # N(50,10^2) から標本抽出
  Xbar = X.mean()
  Sn = X.var()
  rec[i] = (Xbar - 1.96 * ((Sn / n) ** 0.5) < 50) * (50 < Xbar + 1.96 * ((Sn / n) ** 0.5))

rec.mean() # 確率を計算
0.9466

3.4.4 信頼区間の導出

print(stats.norm.cdf(69.6, 50, 10) - stats.norm.cdf(30.4, 50, 10))
0.9500042097035591