Applied Statistics and Data Mining

第3回

第4回

第5回

第6回

第7回

第8回

第9回

第10回: 中間試験

第11回: 二項分布と幾何分布と負の二項分布

休講（出張）

第12回: ポアソン分布と指数分布とガンマ分布

第13回: 正規分布

第14回: 各種不等式と大数の法則と中心極限定理

1.

1. 
Pr[Z=k] 
= Pr[X=1] * Pr[Y=1|X=1] * Pr[Z=k|Y=1]
+ Pr[X=2] * Pr[Y=1|X=2] * Pr[Z=k|Y=1]
+ Pr[X=3] * Pr[Y=1|X=3] * Pr[Z=k|Y=1]
+ Pr[X=1] * Pr[Y=2|X=1] * Pr[Z=k|Y=2]
+ Pr[X=2] * Pr[Y=2|X=2] * Pr[Z=k|Y=2]
+ Pr[X=3] * Pr[Y=2|X=3] * Pr[Z=k|Y=2]
+ Pr[X=1] * Pr[Y=3|X=1] * Pr[Z=k|Y=3]
+ Pr[X=2] * Pr[Y=3|X=2] * Pr[Z=k|Y=3]
+ Pr[X=3] * Pr[Y=3|X=3] * Pr[Z=k|Y=3]

Pr[Z=1]
= Pr[X=1] * Pr[Y=1|X=1] * Pr[Z=1|Y=1]
+ Pr[X=2] * Pr[Y=1|X=2] * Pr[Z=1|Y=1]
+ Pr[X=1] * Pr[Y=2|X=1] * Pr[Z=1|Y=2]
+ Pr[X=2] * Pr[Y=2|X=2] * Pr[Z=1|Y=2]
+ Pr[X=1] * Pr[Y=3|X=1] * Pr[Z=1|Y=3]
+ Pr[X=2] * Pr[Y=3|X=2] * Pr[Z=1|Y=3]
= (8/10) * (1/10) * (6/10)
+ (2/10) * (6/10) * (6/10)
+ (8/10) * (3/10) * (2/10)
+ (2/10) * (3/10) * (2/10)
+ (8/10) * (6/10) * (1/10)
+ (2/10) * (1/10) * (1/10)
= (48+72+48+12+48+2)/1000
= 230/1000

Pr[Z=2]
= Pr[X=1] * Pr[Y=1|X=1] * Pr[Z=2|Y=1]
+ Pr[X=2] * Pr[Y=1|X=2] * Pr[Z=2|Y=1]
+ Pr[X=1] * Pr[Y=2|X=1] * Pr[Z=2|Y=2]
+ Pr[X=2] * Pr[Y=2|X=2] * Pr[Z=2|Y=2]
+ Pr[X=1] * Pr[Y=3|X=1] * Pr[Z=2|Y=3]
+ Pr[X=2] * Pr[Y=3|X=2] * Pr[Z=2|Y=3]
= (8/10) * (1/10) * (3/10)
+ (2/10) * (6/10) * (3/10)
+ (8/10) * (3/10) * (6/10)
+ (2/10) * (3/10) * (6/10)
+ (8/10) * (6/10) * (3/10)
+ (2/10) * (1/10) * (3/10)
= (24+36+144+36+144+6)/1000
= 390/1000

Pr[Z=3]
= Pr[X=1] * Pr[Y=1|X=1] * Pr[Z=3|Y=1]
+ Pr[X=2] * Pr[Y=1|X=2] * Pr[Z=3|Y=1]
+ Pr[X=1] * Pr[Y=2|X=1] * Pr[Z=3|Y=2]
+ Pr[X=2] * Pr[Y=2|X=2] * Pr[Z=3|Y=2]
+ Pr[X=1] * Pr[Y=3|X=1] * Pr[Z=3|Y=3]
+ Pr[X=2] * Pr[Y=3|X=2] * Pr[Z=3|Y=3]
= (8/10) * (1/10) * (1/10)
+ (2/10) * (6/10) * (1/10)
+ (8/10) * (3/10) * (2/10)
+ (2/10) * (3/10) * (2/10)
+ (8/10) * (6/10) * (6/10)
+ (2/10) * (1/10) * (6/10)
= (8+12+48+12+288+12)/1000
= 380/1000

     | Z=1     | Z=2     | Z=3
Prob | 23/100  | 39/100  | 38/100
     | 0.23    | 0.39    | 0.38

2. 

Pr[X=i|Z=k] 
= Pr[X=i, Y=1|Z=k]+ Pr[X=i, Y=2|Z=k]+ Pr[X=i, Y=3|Z=k]
= Pr[X=i] * Pr[Y=1|X=i] * Pr[Z=k|Y=1] / Pr[Z=k]
+ Pr[X=i] * Pr[Y=2|X=i] * Pr[Z=k|Y=2] / Pr[Z=k]
+ Pr[X=i] * Pr[Y=3|X=i] * Pr[Z=k|Y=3] / Pr[Z=k]

Pr[X=1|Z=1]
= Pr[X=1] * Pr[Y=1|X=1] * Pr[Z=1|Y=1] / Pr[Z=1]
+ Pr[X=1] * Pr[Y=2|X=1] * Pr[Z=1|Y=2] / Pr[Z=1]
+ Pr[X=1] * Pr[Y=3|X=1] * Pr[Z=1|Y=3] / Pr[Z=1]
= (8/10) * (1/10) * (6/10) / Pr[Z=1]
+ (8/10) * (3/10) * (2/10) / Pr[Z=1]
+ (8/10) * (6/10) * (1/10) / Pr[Z=1]
= (48+48+48) / Pr[Z=1] / 1000
= 144 / Pr[Z=1] / 1000

Pr[X=2|Z=1]
= Pr[X=2] * Pr[Y=1|X=2] * Pr[Z=1|Y=1] / Pr[Z=1]
+ Pr[X=2] * Pr[Y=2|X=2] * Pr[Z=1|Y=2] / Pr[Z=1]
+ Pr[X=2] * Pr[Y=3|X=2] * Pr[Z=1|Y=3] / Pr[Z=1]
= (2/10) * (6/10) * (6/10) / Pr[Z=1]
+ (2/10) * (3/10) * (2/10) / Pr[Z=1]
+ (2/10) * (1/10) * (1/10) / Pr[Z=1]
= (72+12+2) / Pr[Z=1] / 1000
= 86 / Pr[Z=1] / 1000

  144 / Pr[Z=1] / 1000 
+  86 / Pr[Z=1] / 1000 = 1

Z=1  | X=1      | X=2
Prob | 144/230  | 86/230
     | 0.626087 | 0.373913

3. 

Pr[Y=j|Z=k] 
= Pr[X=1, Y=j|Z=k]+ Pr[X=2, Y=j|Z=k]
= Pr[X=1] * Pr[Y=j|X=1] * Pr[Z=k|Y=j] / Pr[Z=k]
+ Pr[X=2] * Pr[Y=j|X=2] * Pr[Z=k|Y=j] / Pr[Z=k]

Pr[Y=1|Z=1]
= Pr[X=1] * Pr[Y=1|X=1] * Pr[Z=1|Y=1] / Pr[Z=1]
+ Pr[X=2] * Pr[Y=1|X=2] * Pr[Z=1|Y=1] / Pr[Z=1]
= (8/10) * (1/10) * (6/10) / Pr[Z=1]
+ (2/10) * (6/10) * (6/10) / Pr[Z=1]
= (48+72) / Pr[Z=1] / 1000
= 120 / Pr[Z=1] / 1000

Pr[Y=2|Z=1]
= Pr[X=1] * Pr[Y=2|X=1] * Pr[Z=1|Y=2] / Pr[Z=1]
+ Pr[X=2] * Pr[Y=2|X=2] * Pr[Z=1|Y=2] / Pr[Z=1]
= (8/10) * (3/10) * (2/10) / Pr[Z=1]
+ (2/10) * (3/10) * (2/10) / Pr[Z=1]
= (48+12) / Pr[Z=1] / 1000
= 60 / Pr[Z=1] / 1000

Pr[Y=3|Z=1]
= Pr[X=1] * Pr[Y=3|X=1] * Pr[Z=1|Y=3] / Pr[Z=1]
+ Pr[X=2] * Pr[Y=3|X=2] * Pr[Z=1|Y=3] / Pr[Z=1]
= (8/10) * (6/10) * (1/10) / Pr[Z=1]
+ (2/10) * (1/10) * (1/10) / Pr[Z=1]
= (48+2) / Pr[Z=1] / 1000
= 50 / Pr[Z=1] / 1000

  120 / Pr[Z=1] / 1000 
+  60 / Pr[Z=1] / 1000
+  50 / Pr[Z=1] / 1000 = 1

Z=1  | Y=1     | Y=2    | Y=3
Prob | 120/230 | 60/230 | 50/230

E[Y|Z=1] = (1*12+2*6+3*5)/23
         = 39/23
         = 1.695652

1.

1. 
Pr[Z=k] 
= Pr[X=1] * Pr[Y=1|X=1] * Pr[Z=k|Y=1]
+ Pr[X=2] * Pr[Y=1|X=2] * Pr[Z=k|Y=1]
+ Pr[X=1] * Pr[Y=2|X=1] * Pr[Z=k|Y=2]
+ Pr[X=2] * Pr[Y=2|X=2] * Pr[Z=k|Y=2]
+ Pr[X=1] * Pr[Y=3|X=1] * Pr[Z=k|Y=3]
+ Pr[X=2] * Pr[Y=3|X=2] * Pr[Z=k|Y=3]

Pr[Z=1]
= Pr[X=1] * Pr[Y=1|X=1] * Pr[Z=1|Y=1]
+ Pr[X=2] * Pr[Y=1|X=2] * Pr[Z=1|Y=1]
+ Pr[X=3] * Pr[Y=1|X=3] * Pr[Z=1|Y=1]
+ Pr[X=1] * Pr[Y=2|X=1] * Pr[Z=1|Y=2]
+ Pr[X=2] * Pr[Y=2|X=2] * Pr[Z=1|Y=2]
+ Pr[X=3] * Pr[Y=2|X=3] * Pr[Z=1|Y=2]
+ Pr[X=1] * Pr[Y=3|X=1] * Pr[Z=1|Y=3]
+ Pr[X=2] * Pr[Y=3|X=2] * Pr[Z=1|Y=3]
+ Pr[X=3] * Pr[Y=3|X=3] * Pr[Z=1|Y=3]
= (8/10) * (6/10) * (7/10)
+ (1/10) * (0/10) * (7/10)
+ (1/10) * (0/10) * (7/10)
+ (8/10) * (3/10) * (1/10)
+ (1/10) * (6/10) * (1/10)
+ (1/10) * (0/10) * (1/10)
+ (8/10) * (1/10) * (1/10)
+ (1/10) * (4/10) * (1/10)
+ (1/10) *(10/10) * (1/10)
= (336+0+0+24+6+0+8+4+10)/1000
= 388/1000

Pr[Z=2]
= Pr[X=1] * Pr[Y=1|X=1] * Pr[Z=2|Y=1]
+ Pr[X=2] * Pr[Y=1|X=2] * Pr[Z=2|Y=1]
+ Pr[X=3] * Pr[Y=1|X=3] * Pr[Z=2|Y=1]
+ Pr[X=1] * Pr[Y=2|X=1] * Pr[Z=2|Y=2]
+ Pr[X=2] * Pr[Y=2|X=2] * Pr[Z=2|Y=2]
+ Pr[X=3] * Pr[Y=2|X=3] * Pr[Z=2|Y=2]
+ Pr[X=1] * Pr[Y=3|X=1] * Pr[Z=2|Y=3]
+ Pr[X=2] * Pr[Y=3|X=2] * Pr[Z=2|Y=3]
+ Pr[X=3] * Pr[Y=3|X=3] * Pr[Z=2|Y=3]
= (8/10) * (6/10) * (2/10)
+ (1/10) * (0/10) * (2/10)
+ (1/10) * (0/10) * (2/10)
+ (8/10) * (3/10) * (7/10)
+ (1/10) * (6/10) * (7/10)
+ (1/10) * (0/10) * (7/10)
+ (8/10) * (1/10) * (2/10)
+ (1/10) * (4/10) * (2/10)
+ (1/10) *(10/10) * (2/10)
= (96+0+0+168+42+0+16+8+20)/1000
= 350/1000

Pr[Z=3]
= Pr[X=1] * Pr[Y=1|X=1] * Pr[Z=3|Y=1]
+ Pr[X=2] * Pr[Y=1|X=2] * Pr[Z=3|Y=1]
+ Pr[X=3] * Pr[Y=1|X=3] * Pr[Z=3|Y=1]
+ Pr[X=1] * Pr[Y=2|X=1] * Pr[Z=3|Y=2]
+ Pr[X=2] * Pr[Y=2|X=2] * Pr[Z=3|Y=2]
+ Pr[X=3] * Pr[Y=2|X=3] * Pr[Z=3|Y=2]
+ Pr[X=1] * Pr[Y=3|X=1] * Pr[Z=3|Y=3]
+ Pr[X=2] * Pr[Y=3|X=2] * Pr[Z=3|Y=3]
+ Pr[X=3] * Pr[Y=3|X=3] * Pr[Z=3|Y=3]
= (8/10) * (6/10) * (1/10)
+ (1/10) * (0/10) * (1/10)
+ (1/10) * (0/10) * (1/10)
+ (8/10) * (3/10) * (2/10)
+ (1/10) * (6/10) * (2/10)
+ (1/10) * (0/10) * (2/10)
+ (8/10) * (1/10) * (7/10)
+ (1/10) * (4/10) * (7/10)
+ (1/10) *(10/10) * (7/10)
= (1000-388-350)/1000
= 262/1000

     | Z=1      | Z=2      | Z=3
Prob | 388/1000 | 350/1000 | 262/1000
     | 0.388    | 0.350    | 0.262

2. 

Pr[X=i|Z=k] 
= Pr[X=i, Y=1|Z=k]+ Pr[X=i, Y=2|Z=k]+ Pr[X=i, Y=3|Z=k]
= Pr[X=i] * Pr[Y=1|X=i] * Pr[Z=k|Y=1] / Pr[Z=k]
+ Pr[X=i] * Pr[Y=2|X=i] * Pr[Z=k|Y=2] / Pr[Z=k]
+ Pr[X=i] * Pr[Y=3|X=i] * Pr[Z=k|Y=3] / Pr[Z=k]

Pr[X=1|Z=1]
= Pr[X=1] * Pr[Y=1|X=1] * Pr[Z=1|Y=1] / Pr[Z=1]
+ Pr[X=1] * Pr[Y=2|X=1] * Pr[Z=1|Y=2] / Pr[Z=1]
+ Pr[X=1] * Pr[Y=3|X=1] * Pr[Z=1|Y=3] / Pr[Z=1]
= (8/10) * (6/10) * (7/10) / Pr[Z=1]
+ (8/10) * (0/10) * (1/10) / Pr[Z=1]
+ (8/10) * (0/10) * (1/10) / Pr[Z=1]
= 336 / Pr[Z=1] / 1000

Pr[X=2|Z=1]
= Pr[X=2] * Pr[Y=1|X=2] * Pr[Z=1|Y=1] / Pr[Z=1]
+ Pr[X=2] * Pr[Y=2|X=2] * Pr[Z=1|Y=2] / Pr[Z=1]
+ Pr[X=2] * Pr[Y=3|X=2] * Pr[Z=1|Y=3] / Pr[Z=1]
= (1/10) * (3/10) * (7/10) / Pr[Z=1]
+ (1/10) * (6/10) * (1/10) / Pr[Z=1]
+ (1/10) * (0/10) * (1/10) / Pr[Z=1]
= (21+6) / Pr[Z=1] / 1000
= 27 / Pr[Z=1] / 1000

Pr[X=3|Z=1]
= Pr[X=3] * Pr[Y=1|X=3] * Pr[Z=1|Y=1] / Pr[Z=1]
+ Pr[X=3] * Pr[Y=2|X=3] * Pr[Z=1|Y=2] / Pr[Z=1]
+ Pr[X=3] * Pr[Y=3|X=3] * Pr[Z=1|Y=3] / Pr[Z=1]
= (1/10) * (1/10) * (7/10) / Pr[Z=1]
+ (1/10) * (4/10) * (1/10) / Pr[Z=1]
+ (1/10) *(10/10) * (1/10) / Pr[Z=1]
= (7+4+10) / Pr[Z=1] / 1000
= 21 / Pr[Z=1] / 1000

  336 / Pr[Z=1] / 1000 
+  27 / Pr[Z=1] / 1000
+  21 / Pr[Z=1] / 1000 = 1

Z=1  | X=1      | X=2       | X=3
Prob | 336/384  | 27/384    | 21/384
     | 0.875    | 0.0703125 | 0.0546875

3. 

Pr[Y=j|Z=k] 
= Pr[X=1, Y=j|Z=k]+ Pr[X=2, Y=j|Z=k]
= Pr[X=1] * Pr[Y=j|X=1] * Pr[Z=k|Y=j] / Pr[Z=k]
+ Pr[X=2] * Pr[Y=j|X=2] * Pr[Z=k|Y=j] / Pr[Z=k]
+ Pr[X=3] * Pr[Y=j|X=3] * Pr[Z=k|Y=j] / Pr[Z=k]

Pr[Y=1|Z=1]
= Pr[X=1] * Pr[Y=1|X=1] * Pr[Z=1|Y=1] / Pr[Z=1]
+ Pr[X=2] * Pr[Y=1|X=2] * Pr[Z=1|Y=1] / Pr[Z=1]
+ Pr[X=3] * Pr[Y=1|X=3] * Pr[Z=1|Y=1] / Pr[Z=1]
= (8/10) * (6/10) * (7/10) / Pr[Z=1]
+ (1/10) * (0/10) * (7/10) / Pr[Z=1]
+ (1/10) * (0/10) * (7/10) / Pr[Z=1]
= 336 / Pr[Z=1] / 1000

Pr[Y=2|Z=1]
= Pr[X=1] * Pr[Y=2|X=1] * Pr[Z=1|Y=2] / Pr[Z=1]
+ Pr[X=2] * Pr[Y=2|X=2] * Pr[Z=1|Y=2] / Pr[Z=1]
+ Pr[X=3] * Pr[Y=2|X=3] * Pr[Z=1|Y=2] / Pr[Z=1]
= (8/10) * (3/10) * (1/10) / Pr[Z=1]
+ (1/10) * (6/10) * (1/10) / Pr[Z=1]
+ (1/10) * (0/10) * (1/10) / Pr[Z=1]
= (24+6) / Pr[Z=1] / 1000
= 30 / Pr[Z=1] / 1000

Pr[Y=3|Z=1]
= Pr[X=1] * Pr[Y=3|X=1] * Pr[Z=1|Y=3] / Pr[Z=1]
+ Pr[X=2] * Pr[Y=3|X=2] * Pr[Z=1|Y=3] / Pr[Z=1]
+ Pr[X=3] * Pr[Y=3|X=3] * Pr[Z=1|Y=3] / Pr[Z=1]
= (8/10) * (1/10) * (1/10) / Pr[Z=1]
+ (1/10) * (4/10) * (1/10) / Pr[Z=1]
+ (1/10) *(10/10) * (1/10) / Pr[Z=1]
= (8+4+10) / Pr[Z=1] / 1000
= 22 / Pr[Z=1] / 1000

  336 / Pr[Z=1] / 1000 
+  30 / Pr[Z=1] / 1000
+  22 / Pr[Z=1] / 1000 = 1

Z=1  | Y=1     | Y=2    | Y=3
Prob | 336/388 | 30/336 | 22/336

E[Y|Z=1] = (1*336+2*30+3*22)/336
         = 462/336
         = 154/112
         = 77/56
         = 11/8
         = 1.375

原点モーメントは、離散分布なら

<jsmath> m_k=E\left[X^k\right]=\int_{-\infty}^{\infty} x^k p\left(x\right)dx </jsmath>

連続分布なら

<jsmath> m_k=E\left[X^k\right]=\sum_{x=-\infty}^{\infty} x^k p\left(x\right) </jsmath>

で計算する。

中心モーメントは、1次の中心モーメントは

<jsmath> \mu=\mu_1=m_1-m_1=0 </jsmath>

2次の中心モーメント(分散)は

<jsmath> \sigma^2=\mu_2=m_2-{m_1}^2 </jsmath>

3次の中心モーメントは

<jsmath> \mu_3=m_3-3m_2m_1+2{m_1}^3 </jsmath>

など。

<jsmath> M_X\left(t\right) = E\left[\exp\left(tX\right)\right] </jsmath>

が与えられていれば、

<jsmath> m_k = \left.\frac{d^k}{dt^k}M_X\left(t\right)\right|_{t=0} </jsmath>

で求めることができる。代入して不定になるなら、

<jsmath> m_k = \lim_{t\rightarrow 0}\frac{d^k}{dt^k}M_X\left(t\right) </jsmath>

をロピタルの定理を用いて求める。ロピタルの定理は

<jsmath> \lim_{t\rightarrow 0}{a\left(t\right)} = \lim_{t\rightarrow 0}{b\left(t\right)} = 0 \,\, \pm \infty </jsmath>

のときに、

<jsmath> \lim_{t\rightarrow 0}\frac{a^\prime \left(t\right)}{b^\prime \left(t\right)} </jsmath>

が有限の値に収束するなら、

<jsmath> \lim_{t\rightarrow 0}\frac{a\left(t\right)}{b\left(t\right)} = \lim_{t\rightarrow 0}\frac{a^\prime \left(t\right)}{b^\prime \left(t\right)} </jsmath>

となる、という定理である。

モーメント母関数からも確認できる関係。

• 互いに独立に同一のベルヌーイ分布に従うn個の確率変数の和の分布は二項分布に従う(ベルヌーイ分布の和は二項分布)
• 互いに独立に同一の幾何分布に従うn個の確率変数の和の分布は負の二項分布に従う(幾何分布の和は負の二項分布)
• 互いに独立に同一の指数分布に従うn個の確率変数の和の分布はアーラン分布に従う(指数分布の和はアーラン分布)
• 互いに独立に相異なる正規分布に従うn個の確率変数でも、その和の分布は正規分布に従う(正規分布の和は正規分布)
• アーラン分布はガンマ分布と同等
• χ2乗分布はガンマ分布と同等
• 発生間隔が指数分布に従う事象の、一定期間の発生回数はポアソン分布に従う

他にも関係はあるけど、とりあえずこれぐらい。