MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C80CF9.D4650780" このドキュメントは単一ファイル Web ページ (Web アーカイブ ファイル) です。お使いのブラウザ、またはエディタは Web アーカイブ ファイルをサポートしていません。Microsoft Internet Explorer など、Web アーカイブをサポートするブラウザをダウンロードしてください。 ------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布

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マスタ テキストの書式設定
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マスタ タイトルの書式設定
マスタ サブタイトルの書式設定
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<フッター>
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<= /div>
<= /div>
<= /div>
<ヘッダー>= ;
<= /div>
<= /div>
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<日付/時刻>
<= /div>
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------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0001.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
データ分布と予測
確率変数
確率分布
堀田　敬介
2007/10/12,Fri.
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試行とは？
l試行
l何かの行為により「偶然による」ひとつの結果を導き= 出す
さいころ投げ
コイン投げ
l 〔例〕
= l 〔例〕 身長の測定，じゃ= けん，宝くじを買う，
l アンケート調査，製品品質検査，etc.
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file:///C:/8ECB5C12/07dist_2.files/slide0004.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
確率変数とは= ？
<= span class=3DBB style=3D'position:absolute;left:-4.43%;top:.39em'>l確率変数 random variable
lそれがとる値に対し確率が与えられている変数
l=
<= span class=3DB2B style=3D'position:absolute;left:-4.11%;top:.39em'>l例：さいころ投げ
3D"ホームベ=
試行結果
1
2
3
4
5
6
3D"ホームベ=
確率変= 数
の値
1/6
1/6
1/6
1/6
1/6
1/6
3D"ホームベ=
確率
試行してみないと何が出るか= はわからない！
とりうる値はわかっている．
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確率変数とは= ？
=
<= span style=3D'mso-special-format:bullet;color:#CCCC00;mso-color-index:4;positio= n: absolute;left:-3.46%;top:.39em;font-family:Wingdings;font-size:70%'>l例：コイン投げ
3D"ホームベース:
試行結果
表
裏
3D"ホームベ=
確率変数の値
1/2
1/2
3D"ホームベ=
確率
=
<= span style=3D'mso-special-format:bullet;color:#669999;mso-color-index:5;positio= n: absolute;left:-3.17%;top:.39em;font-family:Wingdings;font-size:70%'>l一般に，確率変数の確率は以下のように表現される
ただし，<= /div>
である．<= /div>
= 500
確率はすべて0以上
全ての確率を足すと1
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R0lGODlhrwAqAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQCt ACcAhAAAAAAA/9z//9X//9H//87//8z//9b//+T//+v///L///f///v///7//+P//+r///H///b/ //r///3//////wECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwECAwX/ICCOZGmeaFoGrOq+ cCzPdG3fZCDsLe7/wKAQGBgYj4Ghcsls1liEqDSadFqvWGGgwO16C9WseEw2sQzotDrdK7vfzO13 /g3D7/jndM+f2vOAgSsBB4WGh4iJf4KMdzo7kJGSk5CLjZdiAQibnJ2en55tmKNOLAmnqKmqq6wJ lqSwPgEKtLW2t7i5tq+xvTIBC8HCw8TFxsW8vsonLAzOz9DR0tPUosvXKw3a29zd3t/g3MnYSizm 5+jp6uYU7e7v8PHy8/Hr9vf4+fr7/Pfh/wADChxIsKDBgwgNNqPGsKHDhxAjSpxIsaLFasCOadzI saPHjyBDihxJ8hgLXShT/6pcybKly5cwY8pEaaqVzZs4c+rcybOnz59AW7EARbSo0aNIkypdyrSp 06ebWFCaSrWq1atYs2rdyrWrVx6EEokdS7as2bNo06pdy7ZtWih94sqdS7eu3bt48+rdy5cACzqA AwseTLiw4cOIEytezOXMmseQI0ueTLmy5cuYM2veLPkv48+gQ4seTTow3L6oU6tezbp1XBZHYsue Tbu27du4c+vezbu3799IHn0dTry48ePIp7JwwLy58+fQo0ufTr269evYs2vfzp3Fg+/gw4sfT768 +fPo06tfz769+/csIMifT7++/fv48+vfz7+///8ABiggCxEUaOCBCCao4EqCDDbo4IMQRijhhBRW yIIEGGao4YYcdujhhyCGKOKIJJZo4okosjDBiiy26OKLMMYo44w01mjjjTjmqOOO/fTo449ABink kD6GAAA7 ------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0079.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
演習１
l<= /span>確率変数 <= /div>
l2個のさいころA, Bを振り出た目の差（Aの目ーBの目）を考える．この確率変数 X <= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes;color:#9966FF'>のとる値と，その値が出る= 確率を求めよ．
l<= /span>例）Aが1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes'>で，Bが3の時，1-3 =3D -2
2/36
4<= /div>
3/36
3<= /div>
6/36
0<= /div>
5/36
-1=
4/36
-2=
3/36
-3=
1/36
4/36
5/36
2/36
1/36
P<= /span>(<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>X<= /span>)<= /div>
5<= /div>
2<= /div>
1<= /div>
-4=
-5=
X<= /span>
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確率分布 probability distribution
l確率分布
l例：さいころを1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>回投げる
一様分布
1/6
1/6
1/6
1/6
1/6
1/6
P<= /span>(<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";font-size:85%;mso-fareast-language: JA'>X)
6<= /div>
5<= /div>
4<= /div>
3<= /div>
2<= /div>
1<= /div>
X<= /span>
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Content-Type: image/x-wmz H4sIAAAAAAACC7t+9tgsBjBQYFRgYuYEsYRiGRmADOZAsLgQmGQFYk4mGIuJkRHKYmT6//8/mKXH KAEV44ar42FSYFJgBJmgxsbPIMXwH6SYQQDIPwBkLQPiB0D2MaBh3FA1PAy+iSUZIZUFqQwMAQxc UFEusE5GsEuYwSQAlT9HwLgAAAA= ------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0005_image046.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCAAOAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAB AAEAgAAAAAECAwICRAEAOw== ------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0021.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
確率分布 probability distribution
l確率分布
l= 例：さいころを2回投げたときの出た目の和
三角分布
実は
二項分布<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; font-size:111%;mso-special-format:lastCR;display:none'>
1/36
1/18
1/12
1/9
5/36
1/6
5/36
1/9
1/12
1/18
1/36
P(X)
12
11
10
9
8
7
6
5
4
3
2
X
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D/tkxzZZA0makpQ1iAAAOw== ------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0020.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
確率分布 probability distribution
l離散（型）確率分布 discrete distribution
l可算集合 {x1,<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>x<= /span>2,<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-hint:yes;mso-fareast-l= anguage: JA'>…}の中の値を取る確率変数 X は離 discrete type <= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes'>といわれる．このとき，それぞれの値の確率 <= /div>
l
l を X の確率分布 probability distribution という．
l ただし，
確率分布
probability distribution
一般的な定義<= span style=3D'font-family:"AR P明朝体U";mso-fareast-font-family:"AR P明朝体U";m= so-hansi-font-family: "ＭＳ Ｐ明朝";font-size:156%;text-shadow:auto;mso-special-format:lastCR;di= splay: none'>
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=
<= span style=3D'font-family:"Times New Roman";mso-fareast-hint:yes'>l確率変数Xの期待値
確率変数の期= 待値・分散
<= span class=3DBB style=3D'position:absolute;left:-4.43%;top:.39em'>l期待値 expectati= on, expected value
l確率変数 X の期待値
l= 例：コインを3回投げて表が出る回数の確率分布= 3;
l
l
<= span style=3D'font-family:"Times New Roman";mso-fareast-hint:yes'>l期待値
コインを3回投げると，平均して1.5回表が出ることが期待される
1/8
3/8
3/8
1/8
P<= /span>(<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";font-size:85%;mso-fareast-language: JA'>X)
3<= /div>
2<= /div>
1<= /div>
0<= /div>
X<= /span>
期待値は算術平= ﾏを計算しているのと同じ
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演習２
l期待値を求めよう
l宝くじの期待値
H18年オータムジャンボ宝くじ
（新市町村振興 第511回全国自治宝くじ） 13
　〔1千万枚あたりの当たり本数= 〕
　1等= 　1億5000万円 ×2本
　前後賞　　2500万円 ×4本
　組違賞     　  10万円 ×198本
　2等= 　=      1000万円 ×2本
　3等=     　    100万円 ×20本
　4等=        　     5万円 ×3000本
　5等=     　        1万円 ×20,000本
　6等=        　   3000<= /span>円= ×100,000本
　7等=         = ;  　  300<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; font-size:133%'>円= ×1,000,000本
宝くじに関する洒落
LOTTERY: a ta= x on people who are bad at math
------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0074_image066.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhrwA4AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACt ADYAgAAAAGZmmQKvhI+py+0Po5y0WhDC3bz7D4biSJbmlZ3qyrbuC8dbKtf2jec6S+/+DwwKYb2h 8YhMKg3FpfMJjZaa0qr1ikVQs9yuN7j9isdknqaMTqtB4bX7DWee4/S6um3P66X4vf9v1Ac4SIgj WIiYaKbI2Phy6BgpSQE5aXmpUIm5aanJ+dnoCTpaKEp66meKulqnyvrq5go7W5ahcYuRi7ur28v7 6xsMPCxcTHxsnIy8rNxbAAA7 ------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0023.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
確率変数の期= 待値・分散
l補足：期待値の基本法則
lスカラー倍の期待値
l例：さいころを振って出た目の1000倍円貰える賭
証明：
「確率変数のスカラー倍」の期待値は，
「元の確率変数の期待値のスカラー倍」に等しい
一致する
1/6
5/36
1/9
1/12
1/18
1/36
P(X)
6000
5000
4000
3000
2000
1000
a X
1/6
1/6
1/6
1/6
1/6
1/6
P(X)
6
5
4
3
2
1
X
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データ分布と予測　2．確率変数・確率分布
=
<= span style=3D'font-family:"Times New Roman";mso-fareast-hint:yes'>l確率変数Xの分散
確率変数の期= 待値・分散
l分散 variance&= #13;
l確率変数 X<= /span> <= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>の分散
§
§
l= 例：コインを3回投げて表が出る回数の分布の分散= ﾍ?<= /span>
3
2<= /div>
平均的にどの程度
散らばっているか？
0
1
2
3
分散（ばらつき）
＝平均（期= 待値）からのずれ（の2乗）の平均（期待値）
通常の分散を計算しているのと同じ
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確率変数の期= 待値・分散
l分散は何故必要か？
l= 例：確率変数X<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes'>を，さいころを1回振ったときの目，
l 　　確率変数Yを，さいころを2回振ったときの目の平均
l　としたとき，それぞれの期待値を求めよ．
1
2
3
4
5
6
1
2
3
4
5
6
期待値は平均を，分散は散らばりを表す&= #13;
<= span style=3D'font-family:"Times New Roman";mso-fareast-hint:yes'>l例題のそれぞれの分散の値を求めよ．
期待値と分散を知れば分布の目安になる
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rtvxer+iKSicEY1MCzKpXDanjid0J6VqrVdsVnsbJbrkqPiMTqvXI8+oDI/L5+U2ZUzP6/fyzAPP Fyg4yPJHeIio54eQ2OgIR8L4OEmJJVmJ+UjCldkZCOgZOggqWqrIaZp6Rara2sXqGuslS5sEWxt7 i9uqu5va61sKHBw6TNxpfIyZrEzJ3KyJCl35PJ1YbX2InT0qzd24/c0XLn5a7kl+PpeuHsfeXucN Py4/b24fjZ+vD17P7+7vXzyBiN4RpHGQkMGDCwk2FPjwX0R+E/VtSvjpEsY8IpEABNwogkFFcYsW fKTY8c/IYyUlvGH4JQWbmTRr2qxJoQAAOw== ------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0049.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
一致する
確率変数の期= 待値・分散
l補足：分散の基本法則
lスカラー倍の分散
証明：
l例：さいころ1個を振り，出目の1000倍円貰える賭
576円貰える」だったら <= /span>どちらが計算が楽?
「確率変数のスカラー倍」の分散は，「元の確= 率変数の分散のスカラー2乗倍」に等しい
1/6
5/36
1/9
1/12
1/18
1/36
P(X)
6000
5000
4000
3000
2000
1000
a X
1/6
1/6
1/6
1/6
1/6
1/6
P(X)
6
5
4
3
2
1
X
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lJhWskTS8eUZkzKjuKzJiSbOJDd3sunp80bMoFqAEqVh9CiMpEpZ6WwKiinUE0On5pBqlQTWrCO2 cg3h9Susp2LjkC2L6ixaE1XXslXrFsPIsHE7AKobB0FbvFw+7OWboRdgFscGmy1suEQXuoYKAAA7 ------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0063.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
確率変数の期= 待値・分散
l標準偏差 standard deviation
l= 確率変数Xの標準偏差
l
l
l= 例：コインを3回投げて表が出る回数の分布の標準偏差は?<= /span>
1/8
3/8
3/8
1/8
P<= /span>(<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";font-size:85%;mso-fareast-language: JA'>X)
3<= /div>
2<= /div>
1<= /div>
0<= /div>
X<= /span>
標準偏差
＝分散の平方根
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データ分布と予測　2．確率変数・確率分布
尖度の数値の意味
歪度の数値の意味
補足：確率変数の歪度・尖度
l歪度 skewness
l確率変数Xの確率分布の= 非対称性の指標
l歪度＝α<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";font-size:66%;mso-fareast-hint:yes; position:relative;top:.37em;mso-text-raise:-25%;mso-fareast-language:JA'>3= ．ただし，
…右の裾が長い
…左の裾が長い= 3;
…歪みの程度
l尖度 kurtosis coefficient of exc= ess） = ;
l確率変数Xの確率分布の= 尖り具合を表す指標
l尖度＝α<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";font-size:66%;mso-fareast-hint:yes; position:relative;top:.37em;mso-text-raise:-25%;mso-fareast-language:JA'>4= ー3．ただし，
正規分布よ= り丸く鈍い形
正規分布がα4= ＝3
なので，これと比較
正規分布よ= り尖っている
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image/gif R0lGODlhjQA5AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQCM ADcAgQAAADNm/////wECAwL/hI+py+0ZYni02ouzzlKIuYXiSJZA53mgybbuJgXpvL72jaPzXuP+ D4vshqke8IhU6IhDY/LpizGnKqg1J6RqndduUKYNf7zkr/jMLasR0rM7vUbG5lm3/UPP6/f8vj82 Mnc3SFhoODhnsXTI2OjoKNFQ90hZaWkXoXS5ydlJlGkA5jlKajkhWpqqWni66vr6hgo7S6siW4u7 OpnLW7rbC8z5G0xMOVyMfHiczHy33Awd9hxN/XlbjW2dvT01zU3t/Q0dLs5MXo58jk6svg7c7s4L H487T09rfw+br+/K36/qH0Bf1wYGE2jQE8KEwgoyzLXwYSUQDiW+qhHRIiFQHRAyapQGJ5THj0Ui YdBDMhadF39aunwJMyafDQUAADs= ------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0053.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
演習２
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= Coffee Break!
1から100まで足すといくつ？&= #13;
Q2．1から100までの2乗和は？
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Q1．1から100までの和は？
326から579までの和は？
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Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
確率分布
probability distribution
離散（型）分= z discrete distribution
  = ★（離散）一様分布 uniform distribution
  = ★ベルヌーイ分布 Bernoulli distribution
  = ★二項分布 binomial distribution
  = ★ポアソン分布 Poisson distribution
  = ★幾何分布 geometric distribution
  = ★負の二項分布 negative binomial distribution
  = ★超幾何分布 hypergeometric distribution
------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0006.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
離散型分布 discrete distribution
l(離散)一様分布 uniform distribution (of discrete type)
lすべての確率が等しい分布
l例：さいころを1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>回投げる
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l=
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1/6
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P(X)
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離散型分布 discrete distribution
lベルヌーイ分布 Bernoulli distribution
l試行の結果が2通りしかない確率分布
=
<= span style=3D'mso-special-format:bullet;color:#CCCC00;mso-color-index:4;positio= n: absolute;left:-3.46%;top:.44em;font-family:Wingdings;font-size:70%'>l例：コインを1回投げる
<= span style=3D'mso-special-format:bullet;color:#330066;mso-color-index:3;positio= n: absolute;left:-3.44%;top:.3em;font-family:Wingdings;font-size:75%'>§表が確率2/3，裏が確率1/3で出る硬貨
=
lベルヌーイ試行
= l2<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>通りの結果しかない観測があり，これを同条件で独立にn<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>回行うこと．
1-p
p
P(X)
1
0
X
1/3
2/3
P(X)
1
0
X
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Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0008.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
離散型分布 discrete distribution
l二項分布 binomial distribution
lベルヌーイ試行で，一つの結果が起こる回数の確率 <= /div>
l確率p<= /span>をもつ事象がn<= /span>回の施行中x<= /span>回起こる確率
=
<= span style=3D'font-family:"Times New Roman"'>l例：サイコロを<= /span>3<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>回投げて1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>の目がx<= /span>回出る確率は…=
§0<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>回出る確率　…=
§1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>回出る確率　…=
§2<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>回出る確率　…=
§3<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>回出る確率　…=
1が0回出る
3箇所に = ;
0個1を置く
1が1回出る
3箇所に = ;
1個1を置く
<= /div>
<= /div>
<= /div>
<= /span>
P(X)
3
2
1
0
X
確率分布
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Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
離散型分布 discrete distribution
3D"*"回数を増やすと1の目が x 回出る確率は…
3回
7回
10回
20回
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l二項分布
l確率p<= /span>をもつ事象がn<= /span>回の施行中x<= /span>回起こる確率
離散型分布 discrete distribution
=
<= span style=3D'font-family:"Times New Roman"'>l確率分布・期待= l・分散
p<= /div>
1ーp
n回&#= 13;
x<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; font-size:111%;mso-fareast-hint:yes'>回
p
1ーp
…
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Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
= Coffee Break!
パスカルの三角形と= 項係数 <= /span>
の各項の係数
10 =3D 1 + 3 + 6
10 =3D 1 + 2 + 3 + 4
組合せ数の和法則
二項定理，二項係数
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l二項分布の例
l例1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>：製品ラインの不良品抜き取り検査
l不良率p<= /span>=3D0.5%のロットから独立に1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>個ずつランダム抜取り検査をした時に検出される不良品数x = の従う分布
l参考）不良率の期待値
l例2<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>：袋から球を取り出す
l赤玉3<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>，白玉7<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>入っている袋から1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>つ取り出しては戻すという行為をn<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>回行ったとき，赤玉が5<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>回出る確率は?
l例3<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>：サイコロをn<= /span>回投げて偶数の目が出る回数の従う分布
lサイコロを5<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>回投げて偶数が出る回数の確率分布を求めよ
離散型分布 discrete distribution
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演習３
l二項分布を求めよう
l赤玉3個，白玉7個入っている袋から1つ取り出しては戻すという行為を5回行ったとき，赤球が出る回数の確= 率分布（二項分布）を求めてみよう！
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離散型分布 discrete distribution
lポアソン分布 Poisson distribution =
l= ある時間帯の中で，ある事象が何回起きるか？&#= 13;
l= 例：電話番号案内 <= /div>
§= 事象S＝「通話がある」
0
t
1/2に分割
1/4に分割
1/8に分割
1/16に分割
1/nに分割
3D"テキスト=
…
n<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-fareast-font-family:"ＭＳ Ｐゴ= Vック";mso-fareast-hint: yes'>が十分大きければ2回以上S<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes'>が起きる区間が無くなる
P(ある区間でS<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes'>が2回以上起きる= 確率)=3D0とする
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AAwAgQAAAAAA/5nM/wECAwIxhI+Joe0ZBHsUCjkrjVeuAIbiGHYXaHDmyraY6sYsKNcdSubkmmmK ivI5OD3hL5goAAA7 ------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0065.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
離散型分布 discrete distribution
lポアソン分布 Poisson distribution
1/nに分割
S<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-fareast-font-family:"ＭＳ Ｐゴ= Vック";mso-fareast-hint: yes'>が起きる回数は二項分布 Bi (n, p) に従う
0
t
ベルヌーイ= 試行とみなす
ところで，この時間内にSが起きる回数の期待値をλとおくと…&#= 13;
よって，Sがk<= /span>回起きる確率は，&= #13;
（二項分布の期待値より）&= #13;
各区間でSが起きる確率　 ： p
各区間でSが起きない確= 率： q=3D1-p&#= 13;
とする
確率<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>p<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes'>の事象がn<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes'>回の試行の中でS<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes'>回起こる
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離散型分布 discrete distribution
lポアソン分布 Poisson distribution =
l2項分布においてある事象が起こる確率が非常に小= さい場合に適用できる分布
l例：工場の生産ラインでの不良率が1/500のとき，1000個の製品を作ったときx<= /span>個不良品だった
二項分布
ポアソン分布
1000個のうち，平均的に2個不良品
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lポアソン分布
l確率分布
離散型分布 discrete distribution
l期待値・分散
例では…
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データ分布と予測　2．確率変数・確率分布
離散型分布 discrete distribution
l二項分布からポアソン分布へ
ポアソンの小数の法則
Poisson’s law of small numbers
二項分布
ポアソン分布<= /span>
=
l例：工場の生産ラインでの不良率が1/500のとき，1000個の製品を作ったときx<= /span>個不良品だった
<= ![if !ppt]>3D"角丸四角形:
二項分布
<= ![if !ppt]>3D"角丸四角形:
ポアソン分布
np =3Dλ
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gtOoU58WXbo15jMRYgOIMHt2hMquc5MOIKG379+sdQsPNaG4ceO4hytfHGCD8+fPgy8XHoCD9evW k0/fnqaD9+8dpHPPHcCD+fPax08PME+8+tbs5bl/XzpAwvT0lQcAPz//6AAfBIiff9SBYGB/BGYW QAghDJigbmcg+OBrDk7oWoUWwidhhpZtyAQhY0EAADs= ------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0046.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
離散型分布 discrete distribution
l二項分布とポアソン分布
l例：単一時間に発生する事故件数は?
l一日 m件の事故が発生したとする．これを= 1時間毎，1分毎，1秒毎と縮めていき，1= 刻みに1件の事故が発生するようにし(同時刻に2件発生することはないとする)，その刻み数をnとする．
lすると，この話は n個の刻みの中で 1件事故が発= 生するかしないかとみなすことができる．即ち，n個の刻みの中から事件が発生した x個の刻みの個数を考えることになる
l事故発生率は p =3D m / n =3D 一定！
l (ポアソン分布の期待値)
lこのときの m =3D n p は二項分布の期待値！
------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0037.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
離散型分布 discrete distribution
lポアソン分布の例
l例1：飛行機事故（事故= ﾍめったに起きない）
<= span class=3DB2B style=3D'position:absolute;left:-3.42%;top:.39em'>l飛行機事故の確率1/10= 怐C飛行機搭乗回数を1万回としたとき，一度も事故にあわな= ｢確率は?
l例2：大量生産品の不良= i数（めったにない）
<= span class=3DB2B style=3D'position:absolute;left:-3.49%;top:.39em'>l不良率が1/10000= ﾌ生産ラインで1万個生産したとき不良品が3個以上出る確率は?
l例3：爆撃命中数（めっ= ｽに当たらない）
<= span class=3DB2B style=3D'position:absolute;left:-3.46%;top:.39em'>l第二次大戦中のドイツ軍の砲弾命中精度はλ=3D0.93のポアソン分布に従うという．<= /span>1000発打って1発当たる確率は?
l例4：薬の副作用
l副作用の確率が1/200の薬を5000人が服用したとき，30人以上に副作用が出る確率は?
l例5：生物・植物の= ｶ態・繁茂状況を示す分布
l単位面積あたりのバクテリアの個数
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演習４
lポアソン分布を求めよう
l赤玉１個，白玉99個入っている袋から1つ取り出しては戻すという行為を5回行ったとき，赤球が出る回数の確= 率分布（ポアソン分布）を求めてみよう！
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離散型分布 discrete distribution
l幾何分布 geometric distribution
lベルヌーイ試行において，試行回数を決めずに初めてある事象が起こるまでの試行回数を x = とするときのX=3Dx = の確率分布
幾何数列（等比数列）の形なので，幾何分布とよばれる
=
= <= span style=3D'font-family:"Times New Roman"'>l幾何分布は，時= ﾔを離散的に（1,2,3,…= )<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>考えるとき，初めて何かが起こるまで待つ時間の長さの確率分布　〔（離散的な）待ち時間分布〕
x-1回
x 回目
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file:///C:/8ECB5C12/07dist_2.files/slide0039.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
離散型分布 discrete distribution
l幾何分布
l確率分布
l期待値・分散
例では…
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5x4rJhT5sZsgUY+6k54YnTOEcEjIbxy7sXy5AxGwMsmu3Cl5kWVLksGGoRK2UhbOn4d4AT0aKmaq VZ0gSXX1MdBIkkCoQU35qOsnj43w3Oi39ZrXWmdb+lQ5jm2ej1OtvmJX9ipBQHDPlq2rUJauRIRK CR482KZSVtz2/u2o6uLdpLqoRq76KcXhbAIGEFgmamgym37FJYQJapeDAgAMtE2H1aEXxsg2e6Pp ui8iVwcK6rZVGgCBAQNWC8fYEx/tD+bCht44e51ypCjfRXV8vHbzpim/VLcuHbm+7SLDG39Mngz4 8i4SAAA7 ------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0040.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
離散型分布 discrete distribution
l幾何分布の例
l例1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>：災害の到来
lある1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>年に風水害が起こる確率が1/25であるとする．風水害が起こるのは平均何年に1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>回か?
l上記と同じ災害が20= 年以内に起こる確率は?
l例2<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>：袋から…=
l白玉4<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>つ，赤玉1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>つが入っている袋がある．1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>つ取り出して元に戻すという試行を繰り返したとき，10= 回目に初めて赤玉が取り出される確率は?
l例3<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>：ドアを開けられる鍵を見つけよう！
ln<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>個の鍵束を持っている．かぎ束からひとつ鍵を取り出しド= アを開けるとき，何回目で開くか?<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>　ただし，試した鍵は1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>回毎に鍵束に戻すこととする
------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0056.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
演習５
l<= /span>幾何分布を求めよう <= /div>
<= span class=3DB1B style=3D'position:absolute;left:-3.83%;top:.39em'>l白玉4つ，赤玉1つが入っている袋がある．1つ取り出して元に戻すという試行を繰り返したとき，初めて赤玉が取り出される回の確率分布（幾何分= 布）を求めてみよう！　10
l が取り出される確率は=
l どれだけか？<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; color:#9966FF;mso-special-format:lastCR;display:none'>
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離散型分布 discrete distribution
l負の二項分布 negative binomial distribution
lある事象がk回起こるまでのもうひとつの事象の回数をxとしたときのX=3Dx (x=3D0,1,2,…)の従う分布
二項分布= で二項係数に負も認めた場合にこの分布になるので「負」の二項分布とよばれる
=
= <= span style=3D'font-family:"Times New Roman";layout-flow:horizontal'>lk=3D1のときは幾何分布に等しいため幾何分布の一般化となっ= ている
k 回
x 回
…
最後は成功なので，
k+x-1回からxの場所
を決める組合せ数
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nGyqroKwbZasKaedVrarfp+vnL2zirWyiKW5xryjyM3OtIwcjqDLmNAsurHMnnzJgMmnjdjRqNNv yabSqdNVHVnpSlxl7hwNWPVA83pJ/UdL/gLeACiwoD6DCLdMSsiwBMGGENlFnBjiIUWKFi9GzKix dyHHjgk/gjQocqTAkib9oUyZZCXLfwtfInQpUwjNmj5u4tShcyeOnj5pAA1aIyZRJEOPQjKqdEjS pqaYQs0pdSrPqlZ/Ys0qdCvXol9hhiXydKxEsz3KorXndW2utm4dwo07Qi1du3HxutW7li/aC4AD Cx5MeEICADs= ------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0041.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
離散型分布 discrete distribution
l負の二項分布
l確率分布
幾何分布のk倍
l期待値・分散
例では…
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離散型分布 discrete distribution
l負の二項分布の例
l例1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>：災害の到来
lある1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>年に風水害が起こる確率が1/25であるとする．風水害が起こるのは平均何年に1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>回か?
l上記と同じ災害が20= 年以内に起こる確率は?
l例2<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>：袋から…=
l白玉4<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>つ，赤玉1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>つが入っている袋がある．1<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>つ取り出して元に戻すという試行を繰り返したとき，赤玉が3<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>回取り出されるまでに白玉が40= 回取り出される確率は?
l例3<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>：シリーズものコレクター
l12種類のキャラクターが売られている．ただし，箱を開け= るまで中にどれが入っているかはわからない．あるコレ= クターが全てのキャラクターを集めるためには何個買わね= ばならないか?
------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0057.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
演習６
l負の二項分布を求めよう
lお菓子の付録として，6種類のキャラクターがある．ただし，箱を開けるまで中にどれが入っているかはわからない．全てのキャラクターを集めるためには，お菓子を平均何個買わねばならないか?　負の二項分布を求め，計算しよう！
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Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
離散型分布 discrete distribution
l超幾何分布 hypergeometric distribution =
l例：白玉がM<= /span>個，赤玉がN-M個（全部でN<= /span>個）ある．ここからn<= /span>個抜き出したとき，白玉がx<= /span>個入っている確率は?<= span lang=3DEN-US style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:= "Times New Roman"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-language:JA;mso-special-format:lastCR;display:none'>
n= ﾂ取り出す組合せのうち，白玉x個，赤玉n-x個取り出す組合せの確率
M
N-M
白玉：x個
赤玉：n-x個
n個抜き出す
白玉がx個入っている確率は<= /span>…
ただし，xの取り得る範囲は
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離散型分布 discrete distribution
l超幾何分布
l確率分布
l期待値・分散
例では…
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離散型分布 discrete distribution
l超幾何分布の性質
l非復元抽出（とったものを戻さない）の時に現れる分布
l復元抽出の場合，M/N=3D= p とした二項分布となる
lN<= /span>→∞の場合，条件 M/N →= <= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>p = の元で二項分布となる
------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0045.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
離散型分布 discrete distribution
l超幾何分布の例
l例：資源調査「捕獲再捕獲法 capture-recapture method」
l湖の中の魚の個体数推定など
l 湖に何匹の魚（N<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes'>匹）がいるのか知りたい!
l 動く対象の数え= 繧ｰで難しい!
l 再捕獲により度数分布を書いてN<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes'>を推定
l
l例：ある湖の中に生息している対象魚について，200匹を捕獲し標識をつけた．さてしばらく後，湖から魚を10= 匹獲ったとき，標識がついている魚が2<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"'>匹いた．この湖にはこの魚は何匹いる= と推定されるか?<= /span>
標識再捕獲法
(mark-recapture method)
ともいう
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file:///C:/8ECB5C12/07dist_2.files/slide0085.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
= Coffee Break!
階乗・順列・組合せ<= /span> <= /span>
3D"*"異なる5脚の椅子を一列に並べる並べ方は何通り?<= /span>
3D"*"5脚の椅子に3人の学生が座る座り方は何通り?<= /span>
3D"*"<= /span>倉庫に5脚の椅子があります．使うのに必要な分3脚だけ倉庫から取り出します．取り出し方は何通り?<= /span>
それぞれどんな計算になる= のかしら?
factorial・permutation・combination
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データ分布と予測　2．確率変数・確率分布
確率密度関数
p. d. f.
(probability density function)
連続（型）分布continuous distribution
  = ★（連続）一様分布 uniform distribution
&n= bsp; ★正規分布 normal distribution
&n= bsp; ★標準正規分布 standard normal distribution
&n= bsp; ★指数分布 exponential distribution
  = ★ガンマ分布 Gamma distribution 2= 分布, 指数分布） <= /span>
  ★ベータ分布 Beta distribution
------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/slide0022.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" データ分布と予測　2．確率変数・確率分布
l連続（型）分布 continuous distribution
l確率変数 X の取る値が関数 f(x) により, = 以下で与えられている場合，X は連続型の確= 率分布を持つという
l
l ただし，
確率密度関数= p. d. f.
確率密度関数
probability density function
<= span style=3D'font-family:"Times New Roman"'>l累積分布関数 c.d.f., cumulative distribution function=
f<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";font-size:111%;mso-fareast-languag= e: JA'>(x)
a<= span lang=3DEN-US style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:= "Times New Roman"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; font-size:111%;mso-fareast-language:JA;mso-special-format:lastCR;display:n= one'>
b<= span lang=3DEN-US style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:= "Times New Roman"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; font-size:111%;mso-fareast-language:JA;mso-special-format:lastCR;display:n= one'>
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l連続型確率変数の期待値と分散 <= /div>
l= 連続型確率変数 X <= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes'>の期待値 <= /div>
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l= 連続型確率変数 X の分散
確率密度関数= p. d. f.
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確率密度関数 p. d. f.
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charset="shift_jis" データ分布と予測　2．確率変数・確率分布
連続型分布 continuous distribution
l（連続）一様分布 uniform distribution
l確率密度関数
0
1
1
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連続型分布 continuous distribution
l正規分布 normal distribution
l確率密度関数
平均μ，分散σ2
標準偏差σ= (=3D14.52)
平均μ(=3D43.2)
68.3%
95.5%
99.7%
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連続型分布 continuous distribution
l標準正規分布 standard normal distribution
l確率変数の標準化
l平均μ，分散σ2の正規分布に従う確率変数Xについて
<= span style=3D'mso-fareast-hint:yes'>l確率密度関= 数
確率変数Zは，<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; text-shadow:auto;mso-fareast-hint:yes;color:red'>平均0，分= U1の正規分布に従う
標準正規分布
= 1
0=
= 1
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l二項分布から正規分布へ…
l試行回数n<= /span>を大きくすると，二項分布は正規分布に近づく
l
l
l試行回数n<= /span>が一定の時に，確率p<= /span>を0.5に近づけると，二項分布は正規分布に近づく
連続型分布 discrete distribution
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l二項分布から正規分布へ…
l試行回数n= 大きくすると，二項分布は正規分布に近づく
連続型分布 discrete distribution
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l正規分布による二項分布の近似
l例：内閣支持率
l500人の人に内閣を支持するかどうか聞いたところ，275人が指示すると答えた．
連続型分布 discrete distribution
内閣支持率：
l内閣支持率を p (不支持率 q =3D 1-p) とすると，これは二項分布となる． l点推定では内閣支持率は55= ％である．正規分布近似を考えると，
<= span style=3D'font-family:"Times New Roman";visibility:hidden'>l&#= 13;
lより，95%信頼区間における区間推定では，内閣支持率は 　　　　より50.6%〜59.4%
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l<= /span>ポアソン分布から正規分布へ…
連続型分布 discrete distribution
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連続型分布 continuous distribution
l<= /span>指数分布 exponential distribution
lポアソン分布に従って起きる事象の生起間隔を表現
l確率密度関数
l=
l=
l累積分布関数
l=
l=
l期待値・分散
Ex(λ)
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連続型分布 continuous distribution
l<= /span>指数分布 exponential distribution
l例：サービスの待ち時= 間（チケット売場の列）
l 単位時間をn等分 <= /span>
l ある区間kで誰かが購入を終了：Xk= =3D1，そうでない：Xk= =3D0
l チケット販売開始時点：0T
l X1,X2= ,X3= ,X4= ,…はパラメータp=3Dλ/nのベルヌーイ試行に従う
l →　Tの確率分布を求める
0
T
Xk=3D1
Xk-1=3D0
区間N+1で購入終了　= →　T<= span lang=3DEN-US style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:= "Times New Roman"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes;mso-fareast-language:JA'>≒(1/n)N　（nが十分大きい= 時）
任意の t>0に対し，kを，<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>k<= span lang=3DEN-US style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:= "Times New Roman"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes;mso-fareast-language:JA'>≦nt= ＜k+1を満たす整数= とする
N+1は幾何分布に従う
　 →
1/n
累積分布関数
確率密度関数
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連続型分布 continuous distribution
l<= /span>ガンマ分布 Gamma distribution
l
l確率密度関数
l=
l=
lガンマ関数 Gamma function=
Ga(α,λ)
n<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes'>が自然数の時
：自由度 n のχ2= 分布
：指数分布
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連続型分布 continuous distribution
l<= /span>ベータ分布 Beta distribution
l
l確率密度関数
l=
l=
lベータ関数 Beta function<= /span>
Be(α,β)
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<= span lang=3DEN-US style=3D'font-family:"AR P明朝体U";mso-ascii-font-family:"AR = P明朝体U"; mso-fareast-font-family:"AR P明朝体U";font-weight:normal;color:#006600; mso-fareast-language:JA'>Coffee Break!
Monty-Hole Dilemma
確率的直感
3枚の扉の向こうに
•百万ドル（当たり）　
•山羊（はずれ） 　 = .
•山羊（はずれ） 　 = .
が隠されているよ．あなたは扉= を１つだ= け選んでいいのよ．
　ところで，あなたが選ばなかった２つの扉のうち，山羊の扉を開くから= ，それを見た後で，開く扉を変えてもい= いよ．
　さぁ，どうする？
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<= span lang=3DEN-US style=3D'font-family:"AR P明朝体U";mso-ascii-font-family:"AR = P明朝体U"; mso-fareast-font-family:"AR P明朝体U";font-weight:normal;color:#006600; mso-fareast-language:JA'>Coffee Break!
Monty-Hole Dilemma
確率的直感
　どうしても納得いかない人のため，扉の数を増やしてみましょう！
　最初に選ぶ扉が100万もあったらどうかしら？
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　<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-fareast-font-family:"ＭＳ Ｐゴ= Vック";mso-special-format: lastCR;display:none'>
　100万の扉からあなたが1つを選んだ後で，残り99万9999の扉のうち=
山羊（はずれ）の99万9998の扉を開い= て見せます．
　それでもあなたは，最初の選択を変えない？　あなたの最初の選択は
神懸かり的な幸運に恵まれているのかしら？
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参考文献&= #13;
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l東京大学教養学部統計学教室編　「自然科学の統計学」 東京大学出版会= （1992） = ;
l白石修二 「例題で学ぶ<= span style=3D'font-size:67%'> Excel統計入門」 森北出版（2001<= /span>） = ;
l村上雅人 「なるほど統計学」 海鳴社 （2002<= /span>） = ;
l丹慶勝市 「図解雑学 統計解析」 ナツメ社 （2003<= /span>） = ;
lJ.Matousek, J.Nesetril / 根上生也・中本敦浩 訳 <= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-fareast-font-family:"ＭＳ Ｐゴ= Vック";font-size: 67%;mso-fareast-hint:yes'>「離散数学への招待 上」 シュプリンガー・フェアラーク東京（2002<= /span>） = ;
l徳山豪 「工学基礎 離散数学とその応用」= 数理工学社（<= span lang=3DEN-US style=3D'font-size:67%;mso-fareast-hint:yes;mso-fareast-language:JA'>2003<= /span>） = ;
<= span style=3D'font-size:67%'>lB.Schechter / グラベルロード訳 「My Brain is Open」共立出版（2003）
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g_scaleInFrame||(IsWin("PPTSld") && parent.IsFullScrMode() ) ) document.body.scroll=3D"no" _RSW() if( IsWin("PPTSld") && parent.IsFullScrMode() ) FullScrInit(); =09 MakeSldVis(); ChkAutoAdv() if( runAnimations ) { if( document.all("NSPlay") ) document.all("NSPlay").autoStart =3D false; if( sld.filters && sld.filters.revealtrans ) setTimeout( "document.body.start()", sld.filters.revealtrans.duration * = 1000 ); else document.body.start(); } } function MakeSldVis()=20 { var fTrans=3Dg_showAnimation && SldHasTrans() if( fTrans )=09 { if( g_bgSound ) { idx=3Dg_bgSound.indexOf(","); pptSound.src=3Dg_bgSound.substr( 0, idx ); pptSound.loop=3D -(parseInt(g_bgSound.substr(idx+1))); } SlideObj.filters.revealtrans.Apply()=09 } SlideObj.style.visibility=3D"visible" if( fTrans ) SlideObj.filters.revealtrans.Play() } function MakeNotesVis()=20 { if( !IsNts() ) return false=20 SlideObj.style.display=3D"none" nObj =3D document.all.item("NotesObj") parent.SetHasNts(0) if( nObj ) {=20 nObj.style.display=3D"" parent.SetHasNts(1) } return 1 } function ChkAutoAdv() { if(SldHasTrans()) SlideObj.onfilterchange=3DAutoAdv else AutoAdv() } function AutoAdv() { if(!IsWin("PPTSld") || !gUseSldTimings )return var sld=3DGetCurSld() if( (sld.mAdvDelay>0) && !parent.IsFramesMode() ) setTimeout("parent.GoToNextSld()",sld.mAdvDelay) } function GetObj(id) { if(g_supportsPPTHTML) return document.all(id); else return document.getElementById(id); } function SldHasTrans() { return SlideObj.style.filter !=3D ""; } function GetSldId()=20 { var regExp =3D /file:\/\/\//i var pos =3D location.href.search(regExp) if (MHTMLPrefix !=3D "" && pos !=3D -1) sId =3D location.href.substring(pos) else { sId =3D RemoveFilePrefixFromHref(location.href); var regExp =3D /\// var fixedHref =3D sId var pos =3D -1 =09 pos =3D fixedHref.search(regExp) while (pos !=3D -1) { fixedHref =3D fixedHref.replace(regExp, "\\") pos =3D fixedHref.search(regExp) } =09 if (g_fBaseHyperlink =3D=3D true) sId =3D "file:///" + fixedHref; else sId =3D fixedHref.substring(fixedHref.lastIndexOf('\\') + 1) } =09 return sId } function HideMenu() { if( frames["PPTSld"] && PPTSld.document.all.item("ctx= tmenu") && PPTSld.ctxtmenu.style.display!=3D"none" ) { PPTSld.ctxtmenu.styl= e.display=3D'none'; return true } return false } function IsWin( name ) { return window.name =3D=3D name } function IsNts() { return IsWin("PPTNts") } function IsSldOrNts() { return( IsWin("PPTSld")||IsWin("PPTNts") ) } function SupportsPPTAnimation() { return( navigator.platform =3D=3D "Win32"= && navigator.appVersion.indexOf("Windows")>0 ) } function SupportsPPTHTML() { var appVer=3Dnavigator.appVersion, msie=3DappVer.indexOf("MSIE "), ver=3D0 if( msie >=3D 0 ) ver=3DparseFloat( appVer.substring( msie+5, appVer.indexOf(";",msie) ) ) else ver=3DparseInt(appVer) return( ver >=3D 4 && msie >=3D 0 ) } function _RSW() { if( !g_supportsPPTHTML || IsNts() || ( !g_scaleInFrame && (!IsWin("PPTSld") || !parent.IsFullScrMode()) ) ) return var padding=3D0; if( IsWin("PPTSld") && parent.IsFramesMode() ) padding=3D6 cltWidth=3Ddocument.body.clientWidth-padding cltHeight=3Ddocument.body.clientHeight-padding factor=3D(1.0*cltWidth)/g_origW if( cltHeight < g_origH*factor ) factor=3D(1.0*cltHeight)/g_origH newSize =3D g_origSz * factor if( newSize < 1 ) newSize=3D1 s=3DSlideObj.style s.fontSize=3DnewSize+"px" s.posWidth=3Dg_origW*factor s.posHeight=3Dg_origH*factor s.posLeft=3D(cltWidth-s.posWidth+padding)/2 s.posTop=3D(cltHeight-s.posHeight+padding)/2 if( g_scaleHyperlinks ) ScaleHyperlinks( factor ) } function _InitAnimations() { animRuntimeInstalled =3D ''+document.body.localTime !=3D 'undefined'; isFullScreen =3D (window.name =3D=3D "PPTSld") && !parent.IsFramesMode(); g_animUseRuntime =3D g_showAnimation && animRuntimeInstalled && !(isFullSc= reen && parent.IsSldVisited()); if( g_animUseRuntime ) { collSeq =3D document.all.tags("seq"); if( collSeq !=3D null ) { for(ii=3D0;ii numSlds ) gSldJumpIdx =3D numSlds; if ( gSldJumpIdx >=3D 0 ) { if ( gSldJumpIdx =3D=3D 0 ) gSldJumpIdx =3D 1; var jumpTo =3D parseInt(gSldJumpIdx); gSldJump =3D 0; gSldJumpIdx =3D ""; win.GoToSld( parent.GetSldList().mList[jumpTo-1].mSldHref ) } } } function _KDH() { if( event.keyCode =3D=3D 8 ) { event.returnValue =3D 0; parent.GoToPrevSld(); } }function DocumentOnClick() { if( IsNts() || parent.HideMenu() ) return; if( ( g_allowAdvOnClick && (window.name=3D=3D"PPTSld") && !parent.IsFrames= Mode() ) || (event && event.keyCode=3D=3D32) ) { =09 if( g_animUseRuntime && g_animMainSequence && g_animMainSequence.cangonex= t ) return; parent.GoToNextSld(); } } var g_supportsPPTHTML =3D SupportsPPTHTML(), g_scaleInFrame =3D 1, gId=3D""= , g_bgSound=3D"", g_scaleHyperlinks =3D false, g_allowAdvOnClick =3D 1, g_showInBrowser = =3D 0, gLoopCont =3D 0, gUseSldTimings =3D 1; var g_showAnimation =3D g_supportsPPTHTML && SupportsPPTAnimation() && ( (w= indow.name=3D=3D"PPTSld" && !parent.IsFramesMode()) || g_showInBrowser );va= r g_animManager =3D null; var g_animUseRuntime =3D false; var g_animItemsToHide, g_animInteractiveItems, g_animSlideTime; var g_animMainSequence =3D null; var ENDSHOW_MESG=3D"スライド ショーの最後です。 クリックすると終了します。"= , SCREEN_MODE=3D"Frames", gIsEndShow=3D0, NUM_VIS_SLDS=3D67, SCRIPT_HREF=3D= "script.js", FULLSCR_HREF=3D"fullscreen.htm"; var gCurSld =3D gPrevSld =3D 1, g_offset =3D 0, gNtsOpen =3D gHasNts =3D gO= tlTxtExp =3D 0, gHasNarration =3D 0, gOtlOpen =3D true window.gPPTHTML=3DSupportsPPTHTML() var g_fBaseHyperlink =3D false; var gMainDoc=3Dnew Array(new hrefList("slide0001.htm",1,-1,1),new hrefList(= "slide0002.htm",1,-1,1),new hrefList("slide0004.htm",1,-1,1),new hrefList("= slide0019.htm",1,-1,1),new hrefList("slide0079.htm",1,-1,1),new hrefList("s= lide0005.htm",1,-1,1),new hrefList("slide0021.htm",1,-1,1),new hrefList("sl= ide0020.htm",1,-1,1),new hrefList("slide0003.htm",1,-1,1),new hrefList("sli= de0074.htm",1,-1,1),new hrefList("slide0023.htm",1,-1,1),new hrefList("slid= e0024.htm",1,-1,1),new hrefList("slide0062.htm",1,-1,1),new hrefList("slide= 0049.htm",1,-1,1),new hrefList("slide0063.htm",1,-1,1),new hrefList("slide0= 075.htm",1,-1,1),new hrefList("slide0053.htm",1,-1,1),new hrefList("slide00= 47.htm",1,-1,1),new hrefList("slide0026.htm",1,-1,1),new hrefList("slide000= 6.htm",1,-1,1),new hrefList("slide0007.htm",1,-1,1),new hrefList("slide0008= .htm",1,-1,1),new hrefList("slide0031.htm",1,-1,1),new hrefList("slide0032.= htm",1,-1,1),new hrefList("slide0059.htm",1,-1,1),new hrefList("slide0060.h= tm",1,-1,1),new hrefList("slide0033.htm",1,-1,1),new hrefList("slide0054.ht= m",1,-1,1),new hrefList("slide0064.htm",1,-1,1),new hrefList("slide0065.htm= ",1,-1,1),new hrefList("slide0012.htm",1,-1,1),new hrefList("slide0035.htm"= ,1,-1,1),new hrefList("slide0036.htm",1,-1,1),new hrefList("slide0046.htm",= 1,-1,1),new hrefList("slide0037.htm",1,-1,1),new hrefList("slide0055.htm",1= ,-1,1),new hrefList("slide0009.htm",1,-1,1),new hrefList("slide0039.htm",1,= -1,1),new hrefList("slide0040.htm",1,-1,1),new hrefList("slide0056.htm",1,-= 1,1),new hrefList("slide0030.htm",1,-1,1),new hrefList("slide0041.htm",1,-1= ,1),new hrefList("slide0042.htm",1,-1,1),new hrefList("slide0057.htm",1,-1,= 1),new hrefList("slide0029.htm",1,-1,1),new hrefList("slide0043.htm",1,-1,1= ),new hrefList("slide0044.htm",1,-1,1),new hrefList("slide0045.htm",1,-1,1)= ,new hrefList("slide0085.htm",1,-1,1),new hrefList("slide0068.htm",1,-1,1),= new hrefList("slide0022.htm",1,-1,1),new hrefList("slide0073.htm",1,-1,1),n= ew hrefList("slide0084.htm",1,-1,1),new hrefList("slide0069.htm",1,-1,1),ne= w hrefList("slide0013.htm",1,-1,1),new hrefList("slide0015.htm",1,-1,1),new= hrefList("slide0034.htm",1,-1,1),new hrefList("slide0067.htm",1,-1,1),new = hrefList("slide0048.htm",1,-1,1),new hrefList("slide0070.htm",1,-1,1),new h= refList("slide0017.htm",1,-1,1),new hrefList("slide0080.htm",1,-1,1),new hr= efList("slide0081.htm",1,-1,1),new hrefList("slide0082.htm",1,-1,1),new hre= fList("slide0071.htm",1,-1,1),new hrefList("slide0072.htm",1,-1,1),new href= List("slide0066.htm",1,-1,1)); /********************************************* Frameset functions These functions control slide navigation and state of the frameset. **********************************************/ function RemoveFilePrefixFromHref(href) { var regExp =3D /^file:\/\/\//i; return href.replace(regExp, "") } function FullScrInit() { g_allowAdvOnClick =3D GetCurSld().mAdvOnClk document.body.style.backgroundColor=3D"black" document.oncontextmenu=3Dparent._CM; document.onkeydown =3D _KDH; document.ondragstart=3DCancel document.onselectstart=3DCancel self.focus() } function Redirect( frmId ) {=09 var str=3Ddocument.location.hash,idx=3Dstr.indexOf('#'), sId=3DGetSldId() if(idx>=3D0) str=3Dstr.substr(1); if( window.name !=3D frmId && ( sId !=3D str) ) { obj =3D GetObj("Main-File") window.location.href=3Dobj.href+"#"+sId return 1 } return 0 } var MHTMLPrefix =3D CalculateMHTMLPrefix();=20 function CalculateMHTMLPrefix() { if ( document.location.protocol =3D=3D 'mhtml:') {=20 href=3Dnew String(document.location.href)=20 Start=3Dhref.indexOf('!')+1=20 End=3Dhref.lastIndexOf('/')+1=20 if (End < Start)=20 return href.substring(0, Start)=20 else=20 return href.substring(0, End)=20 } return ''; } function GetTags(base,tag) { if(g_supportsPPTHTML) return base.all.tags(tag); else return base.getElementsByTagName(tag); } function UpdNtsPane(){ if(frames["PPTNts"]) PPTNts.location.replace( MHTMLP= refix+GetHrefObj( gCurSld ).mNtsHref ) } function UpdNavPane( sldIndex ){ if(gNavLoaded) PPTNav.UpdNav() } function UpdOtNavPane(){ if(gOtlNavLoaded) PPTOtlNav.UpdOtlNav() } function UpdOtlPane(){ if(gOtlLoaded) PPTOtl.UpdOtl() } function SetHasNts( fVal ) { if( gHasNts !=3D fVal ) { gHasNts=3DfVal UpdNavPane() } } function ToggleOtlText() { gOtlTxtExp=3D!gOtlTxtExp UpdOtlPane() } function ClearMedia() { // Clear any sounds playing before launching another browser window. Other= wise, // in fullscreen mode, you'll continue to hear the sound in the frames mod= e. if (PPTSld.pptSound) PPTSld.pptSound.loop =3D 0; } function FullScreen() {=20 if ( PPTSld.g_animUseRuntime ) PPTSld.document.body.pause(); ClearMedia(); var href =3D ( document.location.protocol =3D=3D 'mhtml:') ? FULLSCR_HREF = : FULLSCR_HREF+"#"+GetHrefObj(gCurSld).mSldHref; if (MHTMLPrefix !=3D "") href =3D RemoveFilePrefixFromHref(href) if(PPTNav.event.ctrlKey) { var w =3D (window.screen.availWidth * 1.0) / 2.0 var h =3D w * (PPTSld.g_origH * 1.0) / PPTSld.g_origW win =3D window.open( MHTMLPrefix+href,null,"toolbar=3D0,resizable=3D1,top= =3D0,left=3D0," + "width=3D"+ w + ",height=3D" + h ); if( win.document.body && PPTSld.g_animUseRuntime ) win.document.body.PPTSldFrameset=3Dwindow; } else { win =3D window.open( MHTMLPrefix+href,null,"fullscreen=3Dyes" ); if( win.document.body && PPTSld.g_animUseRuntime ) win.document.body.PPTSldFrameset=3Dwindow; } } function ToggleVNarration() { rObj=3DPPTSld.document.all("NSPlay") if( rObj && !PPTSld.g_animUseRuntime ) { if( (rObj.playState =3D=3D 1)||(rObj.playState =3D=3D 0) ) rObj.Play() else if( rObj.playState =3D=3D 2 ) rObj.Pause() else return; } else if( PPTSld.g_animUseRuntime ) { narObj =3D PPTSld.document.all("narrationID") if( narObj ) narObj.togglePause() } } function GetCurSldNum() { =20 obj=3DGetHrefObj(gCurSld) if( obj.mOrigVis =3D=3D 1 ) return obj.mSldIdx else =20 return gCurSld } function GetNumSlds() { =20 if( GetHrefObj(gCurSld).mOrigVis =3D=3D 1 ) return GetSldList().mNumVisSlds; else return GetSldList().mList.length } function GetSldNum( href ) { for(ii=3D0; ii 1 ) PopSldList(); else if( !IsFramesMode() ) { if( gLoopCont ) GoToFirst() else EndShow() } } function GoToPrevSld() { ii=3DgCurSld-1 if( ii > 0 ) { obj=3DGetHrefObj(ii) while ( obj && ( obj.mVis =3D=3D 0 ) && ( ii>0 ) ) obj=3DGetHrefObj(--ii) if( ii =3D=3D 0 ) ii=3D1 GoToSldNum(ii) } } function GoToFirst(){ GoToSld( GetHrefObj(1).mSldHref ) } function GoToLast() { ii=3DGetSldList().mList.length if( ii !=3D gCurSld ) GoToSld( GetHrefObj(ii).mSldHref ) } function GoToSldNum( num ) { if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue obj =3D GetHrefObj( num ) obj.mVis=3D1 gPrevSld=3DgCurSld gCurSld =3D num; =09 if (MHTMLPrefix !=3D "") PPTSld.location.replace(MHTMLPrefix+RemoveFilePrefixFromHref(obj.mSldHref= )) else PPTSld.location.replace(obj.mSldHref) =09 if( IsFramesMode() ) { UpdNavPane(); UpdOtlPane(); UpdNtsPane() } } function GoToSld( href ) { if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue GetHrefObj( GetSldNum(href) ).mVis=3D1 if (MHTMLPrefix !=3D "") PPTSld.location.replace(MHTMLPrefix+RemoveFilePrefixFromHref(href)) else PPTSld.location.replace(href) } function SldUpdated( id ) { if( id =3D=3D GetHrefObj(gCurSld).mSldHref ) return gPrevSld=3DgCurSld gCurSld=3DGetSldNum(id) if( IsFramesMode() ) { UpdNavPane(); UpdOtlPane(); UpdNtsPane() } } function PrevSldViewed(){ GoToSld( GetHrefObj(gPrevSld).mSldHref ) } function HasPrevSld() { return ( gIsEndShow || ( gCurSld !=3D 1 && GetHrefO= bj( gCurSld-1 ).mVis =3D=3D 1 )||( GetCurSldNum() > 1 ) ) } function HasNextSld() { return (GetCurSldNum() !=3D GetNumSlds()) } function CloseWindow() { if( HideMenu() ) return; =09 var event =3D PPTSld.event; if( !IsFramesMode() && event && (event.keyCode=3D=3D27 || event.keyCode=3D= =3D32 || event.type=3D=3D"click" ) ) window.close( self ); CatchNumKeys( self, event ); } function Unload() { gIsEndShow=3D0; } function SetupEndShow() { gIsEndShow=3D1; PPTSld.document.body.scroll=3D"no"; PPTSld.document.onkeypress=3DCloseWindow; PPTSld.document.onclick=3DCloseWindow; PPTSld.document.oncontextmenu=3D_CM; } function EndShow() { if( IsFramesMode() ) return if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue doc=3DPPTSld.document var dir =3D doc.body.dir if( dir !=3D "rtl" ) dir =3D "ltr"; doc.open() doc.writeln('


' + ENDSHOW_MESG + '

') doc.close() } function SetSldVisited(){ GetSldList().mList[gCurSld-1].mVisited=3Dtrue } function IsSldVisited(){ return GetSldList().mList[gCurSld-1].mVisited } function hrefList( sldHref, visible, advDelay, advClk ) { this.mSldHref=3D this.mNtsHref =3D sldHref this.mOrigVis=3D this.mVis =3D visible this.mVisited=3D false this.mAdvDelay=3D advDelay this.mAdvOnClk=3D advClk } function SldList(arr,curSld,fEnd) { this.mCurSld =3D curSld; this.mList =3D new Array(); var idx =3D 1; for(ii=3D0;ii 0) { PushSldList(sldList,fEnd); gCurSld =3D 1; } else if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue } function PushSldList(arr,fEnd) { var ii =3D gSldStack.length; gSldStack[ii] =3D new SldList(arr,gCurSld,fEnd); GoToSld( gSldStack[ii].mList[0].mSldHref ); } function PopSldList() { if (gSldStack[gSldStack.length-1].fEndShow) EndShow() else { gCurSld =3D gSldStack[gSldStack.length-1].mCurSld; gSldStack[gSldStack.length-1] =3D null; gSldStack.length--; var sldList =3D gSldStack[gSldStack.length-1]; GoToSld( sldList.mList[gCurSld - 1].mSldHref ); } } var custShowList=3Dnew Array(); /********************************************* Navigation button implementation There are 2 types of buttons: ImgBtn, TxtBtn implemented as function objects. They share a similiar interface so the event handlers can call SetActive, for example, on a button=20 object without needing to know exactly=20 what type of button it is. **********************************************/ //---------------------------------- function ImgBtn( oId,bId,w,action ) //---------------------------------- { var t=3Dthis t.Perform =3D _IBP t.SetActive =3D _IBSetA t.SetInactive=3D _IBSetI t.SetPressed =3D _IBSetP t.SetDisabled=3D _IBSetD t.Enabled =3D _IBSetE t.ChangeIcon =3D null t.UserAction =3D action t.ChgState =3D _IBUI t.mObjId =3D oId t.mBorderId=3D bId t.mWidth =3D w t.mIsOn =3D t.mCurState =3D 0 } function _IBSetA() { if( this.mIsOn ) { obj=3Dthis.ChgState( gHiliteClr,gShadowClr,2 ) obj.style.posTop=3D0 } } function _IBSetI() { if( this.mIsOn ) { obj=3Dthis.ChgState( gFaceClr,gFaceClr,1 ) obj.style.posTop=3D0=20 } } function _IBSetP() { if( this.mIsOn ) { obj=3Dthis.ChgState( gShadowClr,gHiliteClr,2 ) obj.style.posLeft+=3D1; obj.style.posTop+=3D1 } } function _IBSetD() { =20 obj=3Dthis.ChgState( gFaceClr,gFaceClr,0 ) obj.style.posTop=3D0=20 } function _IBSetE( state ) { var t=3Dthis GetObj( t.mBorderId ).style.visibility=3D"visible" if( state !=3D t.mIsOn ) { t.mIsOn=3Dstate if( state ) t.SetInactive() else t.SetDisabled() } } function _IBP() { var t=3Dthis if( t.mIsOn ) { if( t.UserAction !=3D null ) t.UserAction() if( t.ChangeIcon ) { obj=3DGetObj(t.mObjId) if( t.ChangeIcon() ) obj.style.posLeft=3Dobj.style.posLeft+(t.mCurState-4)*t.mWidth else obj.style.posLeft=3Dobj.style.posLeft+(t.mCurState-0)*t.mWidth } t.SetActive() } =20 } function _IBUI( clr1,clr2,nextState ) { var t=3Dthis SetBorder( GetObj( t.mBorderId ),clr1,clr2 ) obj=3DGetObj( t.mObjId ) obj.style.posLeft=3Dobj.style.posLeft+(t.mCurState-nextState)*t.mWidth-obj= .style.posTop t.mCurState=3DnextState return obj } //----------------------------------------- function TxtBtn( oId,oeId,action,chkState ) //----------------------------------------- { var t=3Dthis t.Perform =3D _TBP t.SetActive =3D _TBSetA t.SetInactive=3D _TBSetI t.SetPressed =3D _TBSetP t.SetDisabled=3D _TBSetD t.SetEnabled =3D _TBSetE t.GetState =3D chkState t.UserAction =3D action t.ChgState =3D _TBUI t.mObjId =3D oId t.m_elementsId=3D oeId t.mIsOn =3D 1 } function _TBSetA() { var t=3Dthis if( t.mIsOn && !t.GetState() ) t.ChgState( gHiliteClr,gShadowClr,0,0 ) } function _TBSetI() { var t=3Dthis if( t.mIsOn && !t.GetState() ) t.ChgState( gFaceClr,gFaceClr,0,0 ) } function _TBSetP() { if( this.mIsOn ) this.ChgState( gShadowClr,gHiliteClr,1,1 ) } function _TBSetD() { =20 this.ChgState( gFaceClr,gFaceClr,0,0 ) this.mIsOn =3D 0 } function _TBSetE() { var t=3Dthis if( !t.GetState() ) t.ChgState( gFaceClr,gFaceClr,0,0 ) else t.ChgState( gShadowClr,gHiliteClr,1,1 ) t.mIsOn =3D 1 } function _TBP() { var t=3Dthis if( t.mIsOn ) {=20 if( t.UserAction !=3D null ) t.UserAction() if( !t.GetState ) return if( t.GetState() ) t.SetPressed() else t.SetActive() } =20 } function _TBUI( clr1,clr2,lOffset,tOffset ) { SetBorder( GetObj( this.mObjId ),clr1,clr2 ) Offset( GetObj( this.m_elementsId ),lOffset,tOffset ) } function Offset( obj, top, left ){ obj.style.top=3Dtop; obj.style.left=3Dle= ft } function SetBorder( obj, upperLeft, lowerRight ) { s=3Dobj.style; s.borderStyle =3D "solid" s.borderWidth =3D 1=20 s.borderLeftColor =3D s.borderTopColor =3D upperLeft s.borderBottomColor=3D s.borderRightColor =3D lowerRight } function GetBtnObj(){ return gBtnArr[window.event.srcElement.id] } function BtnOnOver(){ b=3DGetBtnObj(); if( b !=3D null ) b.SetActive() } function BtnOnDown(){ b=3DGetBtnObj(); if( b !=3D null ) b.SetPressed() } function BtnOnOut(){ b=3DGetBtnObj(); if( b !=3D null ) b.SetInactive() } function BtnOnUp() { b=3DGetBtnObj() if( b !=3D null ) b.Perform() else Upd() } function GetNtsState(){ return parent.gNtsOpen } function GetOtlState(){ return parent.gOtlOpen } function GetOtlTxtState(){ return parent.gOtlTxtExp } function NtsBtnSetFlag( fVal ) { s=3Ddocument.all.item( this.m_flagId ).style s.display=3D"none" if( fVal ) s.display=3D"" else s.display=3D"none" } function _BSetA_Border(){ b =3D gBtnArr[this.mObjId]; if( b !=3D null ) b.S= etActive() } function _BSetI_Border(){ b =3D gBtnArr[this.mObjId]; if( b !=3D null ) b.S= etInactive() } function _BSetP_Border(){ b =3D gBtnArr[this.mObjId]; if( b !=3D null ) b.S= etPressed() } function _BSetA_BorderImg() {=20 b =3D gBtnArr[this.mBorderId]=20 if( b !=3D null && this.mIsOn && !b.GetState() ) { obj=3Dthis.ChgState( gHiliteClr,gShadowClr,2 ) obj.style.posTop=3D0 } } function _BSetI_BorderImg() {=20 b =3D gBtnArr[this.mBorderId] if( b !=3D null && this.mIsOn && !b.GetState() ) { obj=3Dthis.ChgState( gFaceClr,gFaceClr,1 ) obj.style.posTop=3D0 } } var gHiliteClr=3D"THREEDHIGHLIGHT",gShadowClr=3D"THREEDSHADOW",gFaceClr=3D"= THREEDFACE" var gBtnArr =3D new Array() gBtnArr["nb_otl"] =3D new TxtBtn( "nb_otl","nb_otlElem",parent.ToggleOtlPan= e,GetOtlState ) gBtnArr["nb_otlElem"] =3D new TxtBtn( "nb_otl","nb_otlElem",parent.ToggleOt= lPane,GetOtlState ) gBtnArr["nb_nts"] =3D new ImgBtn( "nb_nts","nb_ntsBorder",10,parent.ToggleN= tsPane ) gBtnArr["nb_nts"].SetActive =3D _BSetA_BorderImg; gBtnArr["nb_nts"].SetInactive =3D _BSetI_BorderImg; gBtnArr["nb_ntsBorder"] =3D new TxtBtn( "nb_ntsBorder","nb_ntsElem",parent.= ToggleNtsPane,GetNtsState ) gBtnArr["nb_ntsElem"] =3D new TxtBtn( "nb_ntsBorder","nb_ntsElem",parent.To= ggleNtsPane,GetNtsState ) gBtnArr["nb_prevBorder"] =3D gBtnArr["nb_prev"]=3D new ImgBtn( "nb_prev","n= b_prevBorder",30,parent.GoToPrevSld ) gBtnArr["nb_nextBorder"] =3D gBtnArr["nb_next"]=3D new ImgBtn( "nb_next","n= b_nextBorder",30,parent.GoToNextSld ) gBtnArr["nb_sldshw"]=3D new ImgBtn( "nb_sldshw","nb_sldshwBorder",18,parent= .FullScreen ) gBtnArr["nb_sldshwBorder"] =3D new TxtBtn( "nb_sldshw","nb_sldshwBorder",pa= rent.FullScreen,null ) gBtnArr["nb_sldshwBorder"].SetActive =3D _BSetA_Border; gBtnArr["nb_sldshwBorder"].SetInactive =3D _BSetI_Border; gBtnArr["nb_sldshwText"] =3D new TxtBtn( "nb_sldshw","nb_sldshwText",parent= .FullScreen,null ) gBtnArr["nb_sldshwText"].SetActive =3D _BSetA_Border; gBtnArr["nb_sldshwText"].SetInactive =3D _BSetI_Border; gBtnArr["nb_voice"] =3D gBtnArr["nb_voiceBorder"] =3D new ImgBtn( "nb_voice= ","nb_voiceBorder",18,parent.ToggleVNarration ) gBtnArr["nb_otlTxtBorder"] =3D gBtnArr["nb_otlTxt"]=3D new ImgBtn( "nb_otlT= xt","nb_otlTxtBorder",23,parent.ToggleOtlText ) gBtnArr["nb_ntsBorder"].m_flagId=3D "nb_nts" gBtnArr["nb_ntsBorder"].SetFlag =3D NtsBtnSetFlag gBtnArr["nb_otlTxt"].ChangeIcon=3D GetOtlTxtState /********************************************* Context menu implementation _CM() is the function that's hooked up to the oncontextmenu event. Once we're asked to show the menu, we first build it by creating DIVs on-the-fly. Then we position it=20 within the screen area so it doesn't get clipped. Creating the DIVs using createElement() means we don't have to write out any extra HTML into the slide HTML files. **********************************************/ var sNext=3D"次へ",sPrev=3D"前へ",sEnd=3D"スライド ショーの終了",sFont=3D"= lＳ 明朝",sArrow=3D"矢印",sFreeform=3D"フリーフォーム",sRect=3D"四角形",sOv= al=3D"楕円" function ShowMenu() { BuildMenu(); var doc=3DPPTSld.document.body,x=3DPPTSld.event.clientX+doc.scrollLeft,y= =3DPPTSld.event.clientY+doc.scrollTop m =3D PPTSld.document.all.item("ctxtmenu") m.style.pixelLeft=3Dx if( (x+m.scrollWidth > doc.clientWidth)&&(x-m.scrollWidth > 0) ) m.style.pixelLeft=3Dx-m.scrollWidth m.style.pixelTop=3Dy if( (y+m.scrollHeight > doc.clientHeight)&&(y-m.scrollHeight > 0) ) m.style.pixelTop=3Dy-m.scrollHeight m.style.display=3D"" } function _CM() { if( !parent.IsFullScrMode() ) return; if(!PPTSld.event.ctrlKey) { ShowMenu() return false } else HideMenu() } function BuildMenu() { if( PPTSld.document.all.item("ctxtmenu") ) return var mObj=3DCreateItem( PPTSld.document.body ) mObj.id=3D"ctxtmenu" mObj.style.visibility=3D"hidden" var s=3DmObj.style s.position=3D"absolute" s.cursor=3D"default" s.width=3D"120px" SetCMBorder(mObj,"menu","black") var iObj=3DCreateItem( mObj ) SetCMBorder( iObj, "threedhighlight","threedshadow" ) iObj.style.padding=3D2 CreateMenuItem( iObj,sNext,M_GoNextSld,M_True ) CreateMenuItem( iObj,sPrev,M_GoPrevSld,M_HasPrevSld ) =09 CreateSeparator( iObj ) CreateMenuItem( iObj,sEnd,M_End,M_True ) mObj.style.visibility=3D"visible" } function Cancel() { window.event.cancelBubble=3Dtrue; window.event.returnVa= lue=3Dfalse } function Highlight() { ChangeClr("activecaption","threedhighlight") } function Deselect() { ChangeClr("threedface","menutext") } function Perform() { e=3DPPTSld.event.srcElement if( e.type=3D=3D"menuitem" && e.IsActive() ) e.Action() else PPTSld.event.cancelBubble=3Dtrue } function ChangeClr( bg,clr ) { e=3DPPTSld.event.srcElement if( e.type=3D=3D"menuitem" && e.IsActive() ) { e.style.backgroundColor=3Dbg e.style.color=3Dclr } } function M_HasPrevSld() { return( parent.HasPrevSld() ) } function M_GoNextSld() { if( gIsEndShow ) M_End(); else GoToNextSld() } function M_GoPrevSld() { if( gIsEndShow ) { gIsEndShow=3D0; history.back();= PPTSld.event.cancelBubble=3Dtrue; } else GoToPrevSld() } function M_True() { return true } function M_End() { window.close( self ) } function CreateMenuItem( node,text,action,eval ) { var e=3DCreateItem( node ) e.type=3D"menuitem" e.Action=3Daction e.IsActive=3Deval e.innerHTML=3Dtext if( !e.IsActive() ) e.style.color=3D"threedshadow" e.onclick=3DPerform e.onmouseover=3DHighlight e.onmouseout=3DDeselect s=3De.style; 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charset="shift_jis" データ分布と予測　2．確率変数・確率分布 ------=_NextPart_01C80CF9.D4650780 Content-Location: file:///C:/8ECB5C12/07dist_2.files/outline.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis"
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