差分
このページの2つのバージョン間の差分を表示します。
両方とも前のリビジョン前のリビジョン次のリビジョン | 前のリビジョン次のリビジョン両方とも次のリビジョン | ||
prob:2013 [2013/07/11 13:52] – watalu | prob:2013 [2013/07/25 11:54] – [モーメントの求め方] watalu | ||
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- | とりあえず配付資料をアップロード。 | + | とりあえず配付資料をアップロード。今年度はちゃんとしたノートになっているのは第10回目以降のみ。他については[[prob: |
+ | |||
+ | === 配付資料 === | ||
{{: | {{: | ||
行 15: | 行 17: | ||
{{: | {{: | ||
- | 中間試験 | + | 第10回: |
- | {{: | + | {{: |
休講(出張) | 休講(出張) | ||
- | {{: | + | {{: |
+ | |||
+ | {{: | ||
+ | |||
+ | {{: | ||
+ | |||
+ | === 中間試験(4) === | ||
+ | 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)/ | ||
+ | = 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)/ | ||
+ | = 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)/ | ||
+ | = 380/1000 | ||
+ | |||
+ | | Z=1 | Z=2 | Z=3 | ||
+ | Prob | 23/ | ||
+ | | 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/ | ||
+ | | 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)/ | ||
+ | = 39/23 | ||
+ | = 1.695652 | ||
+ | </ | ||
+ | |||
+ | === 中間試験(5) === | ||
+ | 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)/ | ||
+ | = 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)/ | ||
+ | = 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)/ | ||
+ | = 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/ | ||
+ | | 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)/ | ||
+ | = 462/336 | ||
+ | = 154/112 | ||
+ | = 77/56 | ||
+ | = 11/8 | ||
+ | = 1.375 | ||
+ | </ | ||
+ | |||
+ | === 中心モーメントの求め方 === | ||
+ | == 直接計算 == | ||
+ | |||
+ | 原点モーメントは、離散分布なら | ||
+ | |||
+ | < | ||
+ | m_k=E\left[X^k\right]=\int_{-\infty}^{\infty} x^k p\left(x\right)dx | ||
+ | </ | ||
+ | |||
+ | 連続分布なら | ||
+ | |||
+ | < | ||
+ | m_k=E\left[X^k\right]=\sum_{x=-\infty}^{\infty} x^k p\left(x\right) | ||
+ | </ | ||
+ | |||
+ | で計算する。 | ||
+ | |||
+ | 中心モーメントは、1次の中心モーメントは | ||
+ | |||
+ | < | ||
+ | \mu=\mu_1=m_1-m_1=0 | ||
+ | </ | ||
+ | |||
+ | 2次の中心モーメント(分散)は | ||
+ | |||
+ | < | ||
+ | \sigma^2=\mu_2=m_2-{m_1}^2 | ||
+ | </ | ||
+ | |||
+ | 3次の中心モーメントは | ||
+ | |||
+ | < | ||
+ | \mu_3=m_3-3m_2m_1+2{m_1}^3 | ||
+ | </ | ||
+ | |||
+ | など。また中心モーメントを求める際、期待値は自ら計算するにしても原点モーメントではなく | ||
+ | |||
+ | < | ||
+ | E\left[X\left(X-1\right)\right] | ||
+ | </ | ||
+ | |||
+ | を求めて、 | ||
+ | |||
+ | < | ||
+ | \sigma^2=\mu_2=E\left[X\left(X-1\right)\right]+m_1-{m_1}^2 | ||
+ | </ | ||
+ | |||
+ | とした方が都合が計算量が少なくなる確率分布もある。(幾何分布、ポアソン分布) | ||
+ | |||
+ | == モーメント母関数からの計算 == | ||
+ | |||
+ | < | ||
+ | M_X\left(t\right) = E\left[\exp\left(tX\right)\right] | ||
+ | </ | ||
+ | |||
+ | が与えられていれば、 | ||
+ | |||
+ | < | ||
+ | m_k = \left.\frac{d^k}{dt^k}M_X\left(t\right)\right|_{t=0} | ||
+ | </ | ||
+ | |||
+ | で求めることができる。代入して不定になるなら、 | ||
+ | |||
+ | < | ||
+ | m_k = \lim_{t\rightarrow 0}\frac{d^k}{dt^k}M_X\left(t\right) | ||
+ | </ | ||
+ | |||
+ | をロピタルの定理を用いて求める。ロピタルの定理は | ||
+ | |||
+ | < | ||
+ | \lim_{t\rightarrow 0}{a\left(t\right)} = \lim_{t\rightarrow 0}{b\left(t\right)} = 0 \,\, \pm \infty | ||
+ | </ | ||
+ | |||
+ | のときに、 | ||
+ | |||
+ | < | ||
+ | \lim_{t\rightarrow 0}\frac{a^\prime \left(t\right)}{b^\prime \left(t\right)} | ||
+ | </ | ||
+ | |||
+ | が有限の値に収束するなら、 | ||
+ | |||
+ | < | ||
+ | \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)} | ||
+ | </ | ||
+ | |||
+ | となる、という定理である。 | ||
+ | |||
+ | これで求まるのは原点モーメントなので、中心モーメントを求めるのは関係式 | ||
+ | |||
+ | < | ||
+ | \mu=\mu_1=m_1-m_1=0, | ||
+ | </ | ||
+ | |||
+ | などを用いるのは、直接計算と同様。 | ||
+ | |||
+ | == 他の分布からの計算 == | ||
+ | |||
+ | 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。 | ||
+ | - 互いに独立な確率変数の和の期待値は、それぞれの期待値の和 | ||
+ | - 互いに独立な確率変数の和の分散は、それぞれの分散の和 | ||
+ | はたたき込んでおくとよい。 | ||