lecture09.ipynb

{
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  {
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   "metadata": {},
   "source": [
    "# 第９回　推測統計（１）"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e97af6d9-8631-4d88-93ea-b23af2329b8d",
   "metadata": {},
   "source": [
    "今回は、確率の考え方、確率分布の考え方、正規分布、二項分布、ポアソン分布、指数分布　などを扱います。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "24332880-8b47-4c62-ad5e-16ff6ce6aae3",
   "metadata": {},
   "source": [
    "第８回で紹介した内容は、統計学の中でも、「記述統計」とよばれる分野に属します。つまり、データそのものの性質を記述し、要約するための方法を紹介しました。そのために、平均、標準偏差、四分位数といった数値や、度数分布表、ヒストグラム、箱ひげ図といった図表を使いました。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "82105132-63f7-463b-87b8-2209229b3d04",
   "metadata": {},
   "source": [
    "これに対して、今回よりのちの内容は、データそのものではなくデータのもとになっている集団の性質を推測するための方法論である「推測統計」という分野に属します。この場合、データは実験や調査によって得られ、データのもとになっている集団の一部を抜き出したものと考えられます。それで、データのことを標本あるいはサンプルといい、データのもとになっている集団のことを母集団といいます。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a281b0f0-c144-434f-b307-b4624508d2db",
   "metadata": {},
   "source": [
    "母集団から標本を取る際、また、標本の性質を理解するためには、確率の考え方が必要になります。確率は、不確実あるいはランダムな現象にかかわる問題を定量的に扱うためのツールとなります。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5c90459c-5065-4f36-9943-1b88a2a37f8a",
   "metadata": {},
   "outputs": [],
   "source": []
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   "cell_type": "markdown",
   "id": "9ee35379-06a5-4a52-8ed9-a7e92ef61d6f",
   "metadata": {},
   "source": [
    "## 事象と確率"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2a800ae7-7d2a-4b3e-a80c-741939cac507",
   "metadata": {},
   "source": [
    "### 確率が用いられる場面"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5298e7fd-3b41-413e-9b0e-0e7b8b16abd1",
   "metadata": {},
   "source": [
    "過去のデータから、起こりうる結果がある程度予測されるが、１回ごとの結果は偶然に支配され、事前に予測できないという場面がいろいろあります。\n",
    "- くじを引いたら何が出るか。サイコロやコインを投げたら何が出るか。\n",
    "- 買おうとする電気製品の寿命はどれだけか\n",
    "- 経営する店に明日、お客が何人来るか\n",
    "\n",
    "行動に先立って、これらの場面で起こる可能性のある現象を不確実性・ランダム性に基づいて評価する必要があります。  \n",
    "また、起きた結果が、今までのデータでは説明できないようなできごとでないかどうか、検証する必要があります。  \n",
    "こういった評価・検証を行う際のツールとして、確率が用いられます。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ddf34659-a570-418e-9d63-30e108d40f90",
   "metadata": {},
   "source": [
    "### 試行と事象"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f9506cad-da48-4dad-864a-a610e6623c6f",
   "metadata": {},
   "source": [
    "#### 基本的な用語の整理"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "44a80a1a-9019-4a45-bc4c-9eb21eff39b2",
   "metadata": {},
   "source": [
    "- 試行：サイコロを投げる、工業製品を製造する、といった例のように、１回ずつあるいは１個ずつの個別の結果が偶然に左右される実験や観測のことをいいます。\n",
    "- 根元（根本）事象：試行によって起こる個々の結果をいいます。\n",
    "- 事象(eventt)：根元事象の集合のことをいいます。ですから、根元事象も事象に含まれます。\n",
    "- 全事象（または標本空間）：すべての根元事象の集合"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b87cf861-4a78-4274-95cf-dfdc8358705f",
   "metadata": {},
   "source": [
    "事象は根元事象の集合を表すので、事象の集合に関して次のような用語があります。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "45f9a4ea-13c0-41b5-a2a9-0cb32c0253ca",
   "metadata": {},
   "source": [
    "- 和事象：事象$A_1, A_2, ..., A_n$のうち、少なくとも１つが起こるという事象（集合の用語では和集合）$$A_1 \\cup A_2 \\cup...\\cup A_n$$\n",
    "- 積事象：事象$A_1, A_2, ..., A_n$が同時に起こるという事象すなわち$$A_1 \\cap A_2 \\cap ... \\cap A_n$$\n",
    "- 空事象：何も起こらないという事象（空集合に相当）。記号$\\Phi$で表す。\n",
    "- 余事象：全事象の中で$A$に含まれていない根元事象からなる事象（補集合に相当）。$A^c$と表し、$$A \\cup A^c = \\Omega$$ $$A \\cap A^c = \\Phi$$\n",
    "\n",
    "- 互いに排反：事象$A_1, A_2, ..., A_n$の中の任意の２つ$A_i$, $A_j$, $i \\neq j$の積事象が空事象$\\Phi$である場合、これらの事象は互いに排反（集合の用語では互いに素）であるという。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "21920a37-3d1d-413b-a452-77c020921e1d",
   "metadata": {},
   "source": [
    "#### 根元事象を列挙する"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "51b953d1-2ddc-4a1e-a98b-43b8193332ec",
   "metadata": {},
   "source": [
    "ある事象の起こる確率を計算するためのひとつの方法は、起こる事象をすべて数え上げて、その中のどれだけ問題とする事象が含まれるか、割合を計算するというやりかたです。そのためにはまず根元事象を数え上げる必要があります。例えば、「'a', 'b', 'c', 'd', 'e'のいずれかが描かれた５枚のカードから３枚を選ぶ」という試行の結果起こる根元事象をすべて列挙してみましょう。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7042c0c1-3e04-4691-a820-c20ca6d17331",
   "metadata": {},
   "source": [
    "このために、「itertools」というライブラリを使います。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "715a3267-1966-474b-b073-adbbe5f7cd81",
   "metadata": {},
   "source": [
    "関数itertools.combinations()はリストと選ぶ枚数を引数に与えると、すべての組み合わせが含まれるオブジェクトを返します。その中身を見るためには、関数list()を使います。下の例のように、ひとつの組み合わせをタプルとし、組み合わせのリストが得られます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "58326d48-890a-4112-aa3b-b9fab9c775a9",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "('a', 'b', 'c')\n",
      "('a', 'b', 'd')\n",
      "('a', 'b', 'e')\n",
      "('a', 'c', 'd')\n",
      "('a', 'c', 'e')\n",
      "('a', 'd', 'e')\n",
      "('b', 'c', 'd')\n",
      "('b', 'c', 'e')\n",
      "('b', 'd', 'e')\n",
      "('c', 'd', 'e')\n"
     ]
    }
   ],
   "source": [
    "import itertools\n",
    "cards = ['a', 'b', 'c', 'd', 'e']\n",
    "three_cards_combi = list(itertools.combinations(cards, 3))\n",
    "for v in three_cards_combi:\n",
    "    print(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "3546011f-e324-4654-b61c-0a19a51ab3fa",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(three_cards_combi)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9f4dce77-b9ae-4015-96f0-2ca4d1653cb4",
   "metadata": {},
   "source": [
    "このように、１０通りの根元事象（根本事象）が得られました。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f0a17cd4-62ca-4e21-804c-8c898ab37f85",
   "metadata": {
    "tags": []
   },
   "source": [
    "### 確率"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d0e79511-e5fe-496f-89fe-eddba189f255",
   "metadata": {},
   "source": [
    "このように定めた「事象」の起こりやすさを測る尺度が「確率」（probability）です。各々の事象には確率の値が対応しています。  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a4b71545-ebe0-4121-9225-6575df27b12d",
   "metadata": {
    "tags": []
   },
   "source": [
    "確率の値は0から1までの間と定めます。また、すべての「根本事象」の確率の総和を1とします。つまり、確率は次のように定義される関数$P$です。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "855df922-3013-49a7-9976-4c50de62bc6b",
   "metadata": {},
   "source": [
    "- 任意の事象$A$について$0 \\leqq P(A) \\leqq 1$\n",
    "- 全事象$\\Omega$について$P(\\Omega) = 1$\n",
    "- $A_1, A_2, ...$が互いに排反な事象ならば、$P(A_1 \\cup A_2 \\cup ...) = P(A_1) + P(A_2) + ...$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a6f6d8bd-3a95-4400-8b1c-ea7010c404d9",
   "metadata": {},
   "source": [
    "例えば、先程の５枚のカードから３枚を選ぶとき、全部で10とおりあり、すべて同じ起こりやすさだと考えられますので、「\"a, b,c\"の組み合わせ」という事象の確率は$1/10 = 0.1$となります。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "67fbfd07-5dd7-422b-a1e4-e55b2a82540f",
   "metadata": {},
   "source": [
    "また、「'a'のカードを含む組み合わせ」は全部で6とおりありますので、そのような事象の確率は6/10 = 0.6となります。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2031ee74-565d-4054-954b-02e6d85fdcf6",
   "metadata": {},
   "source": [
    "さて、この事象と「'a'のカードを含まない組み合わせ」の事象とは互いに「排反」です。つまり、共通する部分集合が存在しません。  \n",
    "そして、全事象は、「'a'のカードを含む組み合わせ」か「'a'のカードを含まない組み合わせ」かのどちらかです。すなわち、これら２つの事象の確率を足すと1になります（このように、標本空間の中で、「事象Aがおこらない」という事象を「余事象」といいます）。  \n",
    "ですから、「'a'のカードを含まない組み合わせ」の事象（つまり、「'a'のカードを含む組み合わせ」の余事象）の確率は、1 - 0.6 = 0.4となります。  \n",
    "実際、上の例で確かに4とおりありますので、確率は4/10 = 0.4となります。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "621ebdfc-8309-4f95-a461-9d01cd355a28",
   "metadata": {},
   "source": [
    "（参考）算数で場合の数を計算してみましょう。まず、「'a'のカードを含む組み合わせ」の場合の数は、次のように数えます。  \n",
    "「1枚は'a'で固定する。2枚目と3枚目を残り4枚から選ぶ。2枚目には4とおり、3枚目には3とおり考えられるので、4x3とおりだが、2枚目と3枚目の順番は関係ないので、2の順列すなわち2! = 2x1で割る。従って、4×3/2×1 = 6とおり」となります。  \n",
    "ですから、「'a'のカードを含まない組み合わせ」は、10 - 6 = 4 とおり　です。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7c8d9b06-cf67-4f14-8529-a7c76829c6a4",
   "metadata": {},
   "source": [
    "### 確率変数と確率分布"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a21168fc-d0d6-4dd9-98f3-c97a9c2b120b",
   "metadata": {},
   "source": [
    "事象に結び付けられた数値を「確率変数」といいます。例えば、コインを５回投げたときの表の回数は、k回表が出るという事象の確率変数といえます。事象と確率は結びついていますから、確率変数は、確率に結びついています。確率変数と確率との対応関係を、「確率分布」といいます。  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7d4200d6-ae30-4ee2-8ab3-afd6574fdd16",
   "metadata": {},
   "source": [
    "### 確率関数と確率密度関数"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "960354aa-aec6-492d-82b5-de4ad8e845dc",
   "metadata": {},
   "source": [
    "確率変数と確率との関係は、確率変数が離散の場合は確率関数、連続の場合は確率密度関数といいます。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1799eb8b-009b-42be-be7a-8c2140da73cf",
   "metadata": {},
   "source": [
    "- 離散型確率変数$X$のとりうる値を$\\{x_1, x_2, ...\\}$とし、それに対して確率$\\{P(X = x_i) = f(x_i) (i = 1, 2, ...)\\}$が与えられているとき、$f(x_i) (i = 1, 2, ...)$を確率関数（または確率質量関数）といいます。\n",
    "- 確率の公理より、次の条件を満たします。$$0 \\leqq f(x_i) \\leqq 1$$ $$\\sum_{i=1}^\\infty f(x_i) = 1$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3a7195ca-f848-44eb-bbe0-6f2dea3db595",
   "metadata": {},
   "source": [
    "一方、連続的な確率変数$X$については、ある特定の一つの値$X = x$に対する確率を決めることはできません。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "89c44ec2-d03f-4708-8327-e0d2d930fa4b",
   "metadata": {},
   "source": [
    "その代わりに、確率変数$X$が任意の$x$における微小な$\\Delta x$において、$X$が区間$[x, x + \\Delta x]$の値を取る確率が確率密度関数$f(x)$を用いて、次のように定義されます。$$P(x \\leqq X \\leqq x + \\Delta x) \\simeq f(x)\\Delta x$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "06aa69a0-cd7e-479c-b50d-248a569dd5a4",
   "metadata": {},
   "source": [
    "これはすなわち、積分の形で書くと、$$P(a \\leqq X \\leqq b) = \\int_a^b f(x)dx$$（ただし$f(x) \\geqq 0$）ということです。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1def89f1-365e-40e1-b891-5dede2c710a3",
   "metadata": {},
   "source": [
    "$f(x)$についても、確率の公理より、$$\\int_{-\\infty}^\\infty f(x)dx = 1$$を満たさなければいけません。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d6cdb688-3945-4646-a0e1-c31dbb4b539e",
   "metadata": {},
   "source": [
    "さて、確率変数$X$が離散型、連続型、どちらの場合でも、累積分布関数（あるいは分布関数）$F(x)$が、$$F(x) = P(X \\leqq x)$$によって定義されます。とくに連続型の場合には、$$F(x) = P(X \\leqq x) = \\int_{-\\infty}^x f(u)du$$と表されます。  \n",
    "つまり累積分布関数が連続で微分可能のとき、その導関数が確率密度関数となります。  \n",
    "離散型の場合は、$$F(x) = \\sum_{i=1}^{n} f(x_i)\\;ただしx_n = x$$となります。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d401dc34-b819-4388-9cdf-2079534642f5",
   "metadata": {},
   "source": [
    "累積分布関数は、後の例題のように、起こってほしくない事象・稀な事象の予測やコントロール（例：品切れの確率を抑えるために必要な在庫は？）のために使います。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "afd217bd-3246-496c-9c3e-9719a7bc4525",
   "metadata": {},
   "source": [
    "### 期待値"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c820818d-f2aa-46ae-b49e-43681e5adb37",
   "metadata": {},
   "source": [
    "離散型の場合、確率変数$X$の期待値は、$f(x)$を確率関数として次のように定義されます。$$E[X] \\equiv \\sum_i x_if(x_i) = \\mu$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9b8ff9af-3367-4c1d-a46d-af22deb5ede4",
   "metadata": {},
   "source": [
    "一方、連続型の場合、確率変数$X$の期待値は、$f(x)$を確率密度関数として、次のように定義されます。$$E[X] \\equiv \\int_{-\\infty}^\\infty xf(x)dx = \\mu$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ab0c8203-bee4-4fab-93fc-ea8dbf5fe214",
   "metadata": {},
   "source": [
    "ここまでの事項について、次の例で検討しましょう。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cedac18d-41c1-4f0e-9036-824faff7aa84",
   "metadata": {},
   "source": [
    "### 例１：コインを５回投げたときの表の枚数$k$の確率分布と期待値"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1fb14a72-ce1c-4b4d-9bc4-3bbf36222ffb",
   "metadata": {},
   "source": [
    "例えば、コインを５回投げたときの表の枚数$k$の確率分布はどうなるでしょうか？  \n",
    "まず、５回投げたときの出かたをすべて書き出してみましょう。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "00268a78-764b-4c0b-bd2d-bbe0997b3fa7",
   "metadata": {},
   "source": [
    "今度も、itertoolsを使います。今度は、関数itertools.product()を使います。引数として与えられるのは、１回の試行でとりうる値のリストと、試行回数$n$を与えるオプションrepeat=$n$です。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "bb4fac61-f40d-4aab-a236-c7c7895296d8",
   "metadata": {},
   "outputs": [],
   "source": [
    "import itertools\n",
    "coin = ['0','1'] # 表が1, 裏が0\n",
    "five_throws = list(itertools.product(coin,repeat=5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "5ddde47e-b2d5-4849-ab59-daed6e0af66d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('0', '0', '0', '0', '0'),\n",
       " ('0', '0', '0', '0', '1'),\n",
       " ('0', '0', '0', '1', '0'),\n",
       " ('0', '0', '0', '1', '1'),\n",
       " ('0', '0', '1', '0', '0'),\n",
       " ('0', '0', '1', '0', '1'),\n",
       " ('0', '0', '1', '1', '0'),\n",
       " ('0', '0', '1', '1', '1'),\n",
       " ('0', '1', '0', '0', '0'),\n",
       " ('0', '1', '0', '0', '1'),\n",
       " ('0', '1', '0', '1', '0'),\n",
       " ('0', '1', '0', '1', '1'),\n",
       " ('0', '1', '1', '0', '0'),\n",
       " ('0', '1', '1', '0', '1'),\n",
       " ('0', '1', '1', '1', '0'),\n",
       " ('0', '1', '1', '1', '1'),\n",
       " ('1', '0', '0', '0', '0'),\n",
       " ('1', '0', '0', '0', '1'),\n",
       " ('1', '0', '0', '1', '0'),\n",
       " ('1', '0', '0', '1', '1'),\n",
       " ('1', '0', '1', '0', '0'),\n",
       " ('1', '0', '1', '0', '1'),\n",
       " ('1', '0', '1', '1', '0'),\n",
       " ('1', '0', '1', '1', '1'),\n",
       " ('1', '1', '0', '0', '0'),\n",
       " ('1', '1', '0', '0', '1'),\n",
       " ('1', '1', '0', '1', '0'),\n",
       " ('1', '1', '0', '1', '1'),\n",
       " ('1', '1', '1', '0', '0'),\n",
       " ('1', '1', '1', '0', '1'),\n",
       " ('1', '1', '1', '1', '0'),\n",
       " ('1', '1', '1', '1', '1')]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "five_throws"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fd3c332d-f3f3-4c64-8674-ef1974d21295",
   "metadata": {},
   "source": [
    "このように、各事象は５つの要素からなるタプルで表現されています。また、そのタプルのそれぞれの要素は表が'1'、裏が'0'という文字で表現されています。  \n",
    "1投めから5投めまでには、2^5 = 32とおりの表・裏の出かたがあります。確かめてみましょう。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "ca6428fc-082c-4f59-a0aa-503c64aaf5d6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "32"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(five_throws)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c2457b4d-192a-4125-8a8f-09fb666566dc",
   "metadata": {},
   "source": [
    "表と裏の出る確率が1/2ずつだとすると、これらの出かたの確率はすべて、$(1/2)^5 = 1/32 = 0.03125$になります。このとき、確率変数すなわち表の枚数の分布を調べてみましょう。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5010862d-dd28-467b-bd72-27b7f95e17c0",
   "metadata": {},
   "source": [
    "配列（アレイ）を使うので、NumPyのライブラリを使います。また、Matplotlibのライブラリpyplotを、ヒストグラムを描くために使います。　  \n",
    "インライン描画を可能にするために%matplotlib inlineを宣言します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "32a1cec0-6249-439c-a940-9edd8e8677d3",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3df01903-350d-4d6b-b414-16dc8ddeef41",
   "metadata": {},
   "source": [
    "それぞれの事象（５投の目の出かた）ごとに表の枚数を格納するための配列count_topsを作り、初期値として0を与えます。  \n",
    "numpy.zeros()は配列の長さを引数として、要素がすべてゼロの配列を返します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "61ac82f7-6653-4bb6-9a32-b3d83d739621",
   "metadata": {},
   "outputs": [],
   "source": [
    "count_tops = np.zeros(len(five_throws)) # 要素数はこの場合32"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "192b9e14-1602-4525-8859-4cdd0d34eabf",
   "metadata": {},
   "source": [
    "配列five_throwsの各要素である、それぞれの事象をあらわすタプル（５つの要素からなる）について表の数を数えます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "2073ce81-73ef-435f-81b5-09d3c666ac7a",
   "metadata": {},
   "outputs": [],
   "source": [
    "t_index = 0 #それぞれの事象の表の数を入れる配列の先頭の添え字\n",
    "#外側のループでは、それぞれの事象の表の数を、あらかじめゼロが全部入った配列を上書きして、記録します。\n",
    "for t in five_throws: # tは1つのタプルです。\n",
    "    num_tops = 0 # その事象の表の数を初期化します。\n",
    "    #内側のループでは、ひとつの事象の１投１投の裏表を取り出し、表の数を累積します。\n",
    "    for i in t: # i はタプルの1つの要素です\n",
    "        i_int = int(i) #表なら1、裏なら0です。\n",
    "        num_tops = num_tops + i_int\n",
    "    count_tops[t_index] = num_tops\n",
    "    t_index += 1 # 次の要素の添え字"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d2f989df-0a40-4508-87bd-eba3059e2cab",
   "metadata": {},
   "source": [
    "32とおりの事象において、表の出る回数は次のようになります。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "4f8578a2-af47-4881-add9-beb7ba0fa51c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0., 1., 1., 2., 1., 2., 2., 3., 1., 2., 2., 3., 2., 3., 3., 4., 1.,\n",
       "       2., 2., 3., 2., 3., 3., 4., 2., 3., 3., 4., 3., 4., 4., 5.])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "count_tops"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1e1ad542-aa43-494a-93ec-fe432f8550f4",
   "metadata": {},
   "source": [
    "5投して表の出る枚数のヒストグラムを次のようにして書きます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "152312df-439e-495e-aa7c-2a6f359f2ea6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 1.,  5., 10., 10.,  5.,  1.]),\n",
       " array([-0.5,  0.5,  1.5,  2.5,  3.5,  4.5,  5.5]),\n",
       " <BarContainer object of 6 artists>)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "intervals = np.arange(-0.5,6.5,1) # 各階級の中心が0, 1, 2, 3, 4, 5になるように端の値を設定します\n",
    "plt.hist(count_tops,bins=intervals)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0c0c6b75-e6fc-4826-96db-37625e42e251",
   "metadata": {},
   "source": [
    "上のヒストグラムは、全事象において、それぞれの枚数が出る事象の数を表しています。  \n",
    "そこで次に、それぞれの枚数が出る確率のヒストグラムを描きます。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2c88e8aa-9858-4623-8204-c33ec750136f",
   "metadata": {},
   "source": [
    "さらにこの次に「二項分布」のヒストグラムを描くことに備えて、上のプログラムを次のように修正します。  \n",
    "まず、それぞれの根元事象が起きる確率を計算します。  \n",
    "つまり、上の$f(x_i)$の計算です。  \n",
    "コインを投げたとき、表が出るか裏が出るかどちらかです（ベルヌーイ試行といいます）。  \n",
    "表の確率を$p$, 裏の確率を$1-p$とすると、例えば表裏表表裏となる確率は、  \n",
    "$p(1-p)pp(1-p)$  \n",
    "となります。  \n",
    "さらに、表が同じ回数の根元事象ごとに、この確率を加算します。そうすることで、新たな事象ー表の回数が同じ根元事象の集合ーの確率が計算できます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "9bb18e45-a8c2-4293-9f19-ca6c31c4bbfd",
   "metadata": {},
   "outputs": [],
   "source": [
    "weight_all = np.zeros(len(five_throws)) # １つの事象（５回のコイン投げ）ごとの重みを格納する配列の初期化。今回は３２個。\n",
    "weight_events = np.zeros(6) # 同じ回数の表を１事象としたときの事象ごとの重みの和の配列の初期化。表の回数は0回から5回まで。\n",
    "t_index = 0 # 1つの事象ごとの表の枚数や重みを格納する配列の添字の初期化。添字は0からなので0で。\n",
    "p = 0.5 # 表の確率\n",
    "q = 1 - p # 裏の確率\n",
    "# 次のループで、３２個の事象ごとに表の回数を数え、その事象の起こる確率を計算する。\n",
    "for t in five_throws:\n",
    "    num_tops = 0 #表の回数を初期化\n",
    "    weight = 1 # 重みを初期化\n",
    "    # 次のループで、事象のタプルを構成する1投ごとに表の枚数と重みを更新する\n",
    "    for i in t:\n",
    "        i_int = int(i) # 文字列の'1', '0'を整数に変換\n",
    "        num_tops = num_tops + i_int # 表の枚数を更新\n",
    "        weight = weight * p**i_int * q**(1-i_int) # 表ならpを、裏ならqを、重みにかけることで重みを更新する。\n",
    "    count_tops[t_index] = num_tops # ループを抜けたので表の枚数を格納する\n",
    "    weight_all[t_index] = weight # ループを抜けたので重みを格納する\n",
    "    weight_events[num_tops] += weight # 表の回数ごとに重みを足していく\n",
    "    t_index += 1 # 添字を１増やす"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1e8599cf-af55-4fc6-8a41-b66ee5d6a22b",
   "metadata": {},
   "source": [
    "各事象の表の枚数は、さきほどと同じです。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "3d827f63-ec58-4c1b-85e4-39858584b9c9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0., 1., 1., 2., 1., 2., 2., 3., 1., 2., 2., 3., 2., 3., 3., 4., 1.,\n",
       "       2., 2., 3., 2., 3., 3., 4., 2., 3., 3., 4., 3., 4., 4., 5.])"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "count_tops"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "876b3fbd-49d1-42c3-bc3b-d04010b87f9e",
   "metadata": {},
   "source": [
    "各根元事象の重み（確率）はすべて同じで、$(0.5)^5 = 0.03125$です。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "2127de43-2849-43d6-a94d-8a7e1a16cefd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.03125, 0.03125, 0.03125, 0.03125, 0.03125, 0.03125, 0.03125,\n",
       "       0.03125, 0.03125, 0.03125, 0.03125, 0.03125, 0.03125, 0.03125,\n",
       "       0.03125, 0.03125, 0.03125, 0.03125, 0.03125, 0.03125, 0.03125,\n",
       "       0.03125, 0.03125, 0.03125, 0.03125, 0.03125, 0.03125, 0.03125,\n",
       "       0.03125, 0.03125, 0.03125, 0.03125])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "weight_all"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b3d4615e-9852-4cf8-9023-3d5a12f05367",
   "metadata": {},
   "source": [
    "表の枚数が0, 1, 2, 3, 4, 5枚である確率はそれぞれ次のようになります。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "7cdb8fe2-b847-4914-a0c0-35be25e92b87",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.03125, 0.15625, 0.3125 , 0.3125 , 0.15625, 0.03125])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "weight_events"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3a970f8e-32ae-4286-9fac-581720d7ceae",
   "metadata": {},
   "source": [
    "確率ですから、これらの和は当然1にならねばいけません。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "4b36dcd0-3a18-4db2-8795-60d748a357ba",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(weight_events)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "68136298-cea3-4d1f-9ba4-bf605b5142f6",
   "metadata": {},
   "source": [
    "表の枚数の確率をヒストグラムにするには、Matplotlib.pyplot.hist()関数のweightsオプションに、上で求めた、それぞれの事象の重み（確率）を与えればよいです。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "98763a78-69fc-49ae-bb91-c0bc7d4c7c76",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([0.03125, 0.15625, 0.3125 , 0.3125 , 0.15625, 0.03125]),\n",
       " array([-0.5,  0.5,  1.5,  2.5,  3.5,  4.5,  5.5]),\n",
       " <BarContainer object of 6 artists>)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(count_tops, weights=weight_all, bins=intervals)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e5a5a696-311f-40d7-8272-c358dd214f48",
   "metadata": {},
   "source": [
    "表の枚数の期待値が二項分布の期待値$np = 5x0.5$に等しいことを確かめましょう。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3ee06c21-0116-4dd0-a3de-e5737ac68e39",
   "metadata": {},
   "source": [
    "期待値は、それぞれの確率変数の値ｘ確率　の総和です。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "db8e7e20-bb1a-4d05-a40c-af98ddb28a3f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.5"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e_tops = np.dot(count_tops,weight_all)\n",
    "e_tops"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a0666d38-cae5-4bb1-902c-cc14c1b58e54",
   "metadata": {},
   "source": [
    "表の枚数の分散が二項分布の分散npq = 5x0.5x0.5に等しいことを確かめましょう。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "977eb8a0-d0a8-4500-8431-2090ed0fa629",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.25"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# array型（配列）の変数からスカラをひくと、配列のすべての要素からスカラをひきます\n",
    "# array型（配列）の変数どうしの「*」演算は、要素どうしの積です\n",
    "sum((count_tops - e_tops) * (count_tops - e_tops) * weight_all)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5f619841-b9cb-4966-9394-ef0db173a65b",
   "metadata": {},
   "source": [
    "根元事象の起こる確率はすべて同じなので、単純にnp.var()関数で分散を求めることもできます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "2c49b0e9-16de-4357-835a-6700ccf1531c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.25"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.var(count_tops)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6b51b244-b6a7-4620-ac9a-84734392087f",
   "metadata": {},
   "source": [
    "## 二項分布"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eeae3447-ab8d-486f-a0ba-a28863a49ca0",
   "metadata": {},
   "source": [
    "### 例２：表と裏の出る確率が違うコインを投げる"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "45799240-0161-4980-b6a2-b8868a4f3234",
   "metadata": {},
   "source": [
    "仮に、コインの表の出る確率が0.4, 裏の出る確率が1-0.4 = 0.6ならば、5枚のうち表の出る枚数の確率分布はどうなるでしょうか？"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f0ce5b1f-e17b-4cd5-b223-a93057f004ae",
   "metadata": {},
   "source": [
    "さきほどのプログラムで、p = 0.4として実行してみましょう。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "ff287e90-8586-4c0e-9bc7-3810f8d6f34c",
   "metadata": {},
   "outputs": [],
   "source": [
    "weight_all = np.zeros(len(five_throws)) # １つの事象（５回のコイン投げ）ごとの重みを格納する配列の初期化\n",
    "weight_events = np.zeros(6) # 同じ回数の表を１事象としたときの事象ごとの重みの和の配列の初期化。\n",
    "# 表の回数は0回から5回まで。\n",
    "t_index = 0 # 1つの事象ごとの表の枚数や重みを格納する配列の添字の初期化。添字は0からなので0で。\n",
    "p = 0.4 # 表の確率\n",
    "q = 1 - p # 裏の確率\n",
    "for t in five_throws:\n",
    "    num_tops = 0 #表の回数を初期化\n",
    "    weight = 1 # 重みを初期化\n",
    "    # 次のループで、事象のタプルを構成する1投ごとに表の枚数と重みを更新する\n",
    "    for i in t:\n",
    "        i_int = int(i) # 文字列の'1', '0'を整数に変換\n",
    "        num_tops = num_tops + i_int # 表の枚数を更新\n",
    "        weight = weight * p**i_int * q**(1-i_int) # 表ならpを、裏ならqを、重みにかけることで重みを更新する。\n",
    "    count_tops[t_index] = num_tops # ループを抜けたので表の枚数を格納する\n",
    "    weight_all[t_index] = weight # ループを抜けたので重みを格納する\n",
    "    weight_events[num_tops] += weight # 表の回数ごとに重みを足していく\n",
    "    t_index += 1 # 添字を１増やしてループの最初に戻る"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "771ffcf8-f9d0-4db8-8eaf-ba64856ad7f3",
   "metadata": {},
   "source": [
    "今度は、３２とおりの各事象の確率はつぎのようになります。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "ea9ed2b2-62ca-4be7-8284-11819b906b2e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.07776, 0.05184, 0.05184, 0.03456, 0.05184, 0.03456, 0.03456,\n",
       "       0.02304, 0.05184, 0.03456, 0.03456, 0.02304, 0.03456, 0.02304,\n",
       "       0.02304, 0.01536, 0.05184, 0.03456, 0.03456, 0.02304, 0.03456,\n",
       "       0.02304, 0.02304, 0.01536, 0.03456, 0.02304, 0.02304, 0.01536,\n",
       "       0.02304, 0.01536, 0.01536, 0.01024])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "weight_all"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3433816e-306d-4129-a9f2-2df9a3f01038",
   "metadata": {},
   "source": [
    "表が0~5枚の確率は、それぞれ次のようになります。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "f735c8f1-5fb2-419a-a8d9-8b839a2a068c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.07776, 0.2592 , 0.3456 , 0.2304 , 0.0768 , 0.01024])"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "weight_events"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b9bef823-dab5-41a2-9d1e-b0f011c1720e",
   "metadata": {},
   "source": [
    "当然、これらの和は1です。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "73cc3bfb-89a3-4a8d-b717-7598000d9684",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(weight_events)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "df76a757-d16b-4c34-bd08-22f0cf8d740a",
   "metadata": {},
   "source": [
    "表が0~5枚の確率をヒストグラムにしてみましょう。0枚、1枚の確率が5枚、4枚よりも高くなっているのがわかりますね。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "f059c0d8-7803-4a33-b036-4b58066290dc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([0.07776, 0.2592 , 0.3456 , 0.2304 , 0.0768 , 0.01024]),\n",
       " array([-0.5,  0.5,  1.5,  2.5,  3.5,  4.5,  5.5]),\n",
       " <BarContainer object of 6 artists>)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(count_tops, weights=weight_all, bins=intervals)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5135c434-243a-4cd8-92c1-f20c0ff3f400",
   "metadata": {},
   "source": [
    "平均が、二項分布$B(n,p) = B(5,0.4)$ の平均$np = 5 x 0.4 = 2$に等しいことを確かめます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "f9ec9ccb-b988-41ee-abfb-bee7255321a2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.0"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e_tops = np.dot(count_tops,weight_all)\n",
    "e_tops"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4604f3bc-95f8-4626-a847-2285a7d9ea08",
   "metadata": {},
   "source": [
    "分散が、npq = 5 x 0.4 x 0.6に等しいことを確かめます。３とおり（実質２とおり）で計算しましたが、誤差がみられます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "5fdbf9cc-4a34-4066-92c0-159847eed5de",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.1999999999999997"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum((count_tops - e_tops) * (count_tops - e_tops) * weight_all)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "0a5178fb-b7d7-40cf-a58c-77381dd99077",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.2000000000000002"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot((count_tops - e_tops) * (count_tops - e_tops),weight_all)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "0714bbb8-f909-41b7-99b4-6558d21892ef",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.2000000000000002"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot((count_tops - e_tops)**2,weight_all)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4b4c6030-00ae-47e6-9414-01ac63cad504",
   "metadata": {},
   "source": [
    "### 二項分布とは"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7285593f-ed63-4ed8-9f09-273588743233",
   "metadata": {},
   "source": [
    "二項分布とは、次のような確率関数$P(X = x)$をもつ確率分布です。\n",
    "- 成功確率$p$の$n$回のベルヌーイ試行を行ったとき、成功の回数が$x$、失敗の回数が$n - x$とする。\n",
    "- 成功の回数$X = x$に対する確率関数$P(X=x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "909e0fcc-4885-4f18-9116-abce57c73b08",
   "metadata": {},
   "source": [
    "また、このように試行回数$n$と成功確率$p$の２つのパラメータで定義できますので、二項分布を$B(n,p)$と表現します。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b26493c0-3cda-4bfd-81a8-a3bc6d40a6bc",
   "metadata": {},
   "source": [
    "確率関数$P(X = x)$は次のように定義できます。$$P(X=x) \\equiv f(x) = nCxp^x(1 - p)^{n - x} \\;\\; (x = 0, 1, 2, .., n)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3b1b0ed9-7d74-48be-a626-ededc25a8356",
   "metadata": {},
   "source": [
    "二項分布の確率関数の値はScipyのstats.binom(n=$n$,p=$p$).pmf([0,1,2,...n])で得られます。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "42c7fb7f-6e95-4ead-ba6b-bea1945f61c7",
   "metadata": {},
   "source": [
    "### 例３　Scipyに含まれる二項分布の確率関数"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ac39024-25cc-457f-b253-8e89c44e713f",
   "metadata": {},
   "source": [
    "上の例２で得られた確率分布を、二項分布の確率関数と比較してみましょう。  \n",
    "SciPyのstats.binom()関数を使います。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "56bee213-830d-49b3-95e8-0b7a43920d22",
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import stats"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "71cde5fd-7e40-4e8f-a9c9-6a65ae376a7b",
   "metadata": {},
   "source": [
    "離散型の確率分布なので確率密度関数（probability density function）ではなく、「pmf: Probability Mass Function（確率質量関数）」であらわします。確率変数(この場合、0,1,2,3,4,5)ごとの確率が確率質量関数からわかります。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "b10565c3-85b0-4796-ac8b-d56b6a0e062e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.07776, 0.2592 , 0.3456 , 0.2304 , 0.0768 , 0.01024])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.binom(n=5,p=0.4).pmf([0,1,2,3,4,5])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "913e3ed9-5f6c-43c9-917f-090ba92a0219",
   "metadata": {},
   "source": [
    "このように、上で求めた値と一致しました。ヒストグラムに確率質量関数を重ね描きしましょう。  \n",
    "単に値をプロットするには、matplotlib.pyplot.plot()関数を用います。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "e6df627d-4a73-4553-8d92-e3ad5cfa13de",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x15e8a142f70>]"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(count_tops, weights=weight_all, bins=intervals)\n",
    "plt.plot([0,1,2,3,4,5],stats.binom(n=5,p=0.4).pmf([0,1,2,3,4,5]),'o')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c1ed7432-a181-4f58-9b14-3ed25e133a04",
   "metadata": {},
   "source": [
    "## ポワソン分布"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ccf7bd6e-f518-4ec4-b824-9bdc9c4feb85",
   "metadata": {},
   "source": [
    "### 例４　稀な現象の二項分布による分析（ポワソン分布への導入）"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f2365aab-00fe-4195-a845-c8bbfdc48f61",
   "metadata": {},
   "source": [
    "だいたい100年に１回おこるできごとがあるとします。それが100年間に2回とか3回おこる可能性はどれくらいでしょう？  \n",
    "100年間に1回もおこらない可能性はどれくらいでしょう？"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c7db06ec-f2d2-4034-a649-3a0c53711a7f",
   "metadata": {},
   "source": [
    "まずは、確率p=0.01で、100回の試行したとき、確率変数0, 1, 2, 3, ..., 99, 100の確率分布を２項分布B(100, 0.01)で計算しましょう。  \n",
    "そのためにまず、NumPy, SciPyをインポートし、確率変数の配列を作ります。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "7c00f668-9150-4835-8e22-6caa41e3e219",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import stats\n",
    "number_events = np.arange(0,101)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1d0924c1-6166-4f8b-9e30-75e0beb87f90",
   "metadata": {},
   "source": [
    "確率変数の一覧の確認です。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "6678dad4-d7f1-415e-ba6d-3e7045c693b9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,\n",
       "        13,  14,  15,  16,  17,  18,  19,  20,  21,  22,  23,  24,  25,\n",
       "        26,  27,  28,  29,  30,  31,  32,  33,  34,  35,  36,  37,  38,\n",
       "        39,  40,  41,  42,  43,  44,  45,  46,  47,  48,  49,  50,  51,\n",
       "        52,  53,  54,  55,  56,  57,  58,  59,  60,  61,  62,  63,  64,\n",
       "        65,  66,  67,  68,  69,  70,  71,  72,  73,  74,  75,  76,  77,\n",
       "        78,  79,  80,  81,  82,  83,  84,  85,  86,  87,  88,  89,  90,\n",
       "        91,  92,  93,  94,  95,  96,  97,  98,  99, 100])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "number_events"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d50b925f-fd53-4a75-a556-e46561937ee1",
   "metadata": {},
   "source": [
    "それぞれの確率変数に対する、２項分布の確率分布です。離散値をとりますので、pmf()メソッドを用います。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "d8f86691-8ed9-4954-b791-92f0a83f688c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3.66032341e-001, 3.69729638e-001, 1.84864819e-001, 6.09991658e-002,\n",
       "       1.49417149e-002, 2.89778712e-003, 4.63450803e-004, 6.28634566e-005,\n",
       "       7.38169377e-006, 7.62195092e-007, 7.00603569e-008, 5.79011214e-009,\n",
       "       4.33771028e-010, 2.96595574e-011, 1.86174711e-012, 1.07818351e-013,\n",
       "       5.78570698e-015, 2.88769689e-016, 1.34499911e-017, 5.86336668e-019,\n",
       "       2.39865000e-020, 9.23001445e-022, 3.34789321e-023, 1.14684089e-024,\n",
       "       3.71661399e-026, 1.14126329e-027, 3.32535923e-029, 9.20600759e-031,\n",
       "       2.42438151e-032, 6.07995362e-034, 1.45345693e-035, 3.31515102e-037,\n",
       "       7.22049939e-039, 1.50288937e-040, 2.99149103e-042, 5.69807815e-044,\n",
       "       1.03921178e-045, 1.81571264e-047, 3.04066711e-049, 4.88270812e-051,\n",
       "       7.52134332e-053, 1.11180241e-054, 1.57759361e-056, 2.14941107e-058,\n",
       "       2.81259025e-060, 3.53546698e-062, 4.26988766e-064, 4.95538219e-066,\n",
       "       5.52683620e-068, 5.92445851e-070, 6.10398756e-072, 6.04474902e-074,\n",
       "       5.75354899e-076, 5.26339530e-078, 4.62737709e-080, 3.90926255e-082,\n",
       "       3.17310272e-084, 2.47415417e-086, 1.85281486e-088, 1.33227571e-090,\n",
       "       9.19584243e-093, 6.09097031e-095, 3.87011799e-097, 2.35793625e-099,\n",
       "       1.37695141e-101, 7.70322465e-104, 4.12630644e-106, 2.11509753e-108,\n",
       "       1.03681251e-110, 4.85697561e-113, 2.17267307e-115, 9.27303915e-118,\n",
       "       3.77270111e-120, 1.46168024e-122, 5.38702792e-125, 1.88636668e-127,\n",
       "       6.26783187e-130, 1.97334337e-132, 5.87760910e-135, 1.65333589e-137,\n",
       "       4.38384517e-140, 1.09336455e-142, 2.55899563e-145, 5.60568593e-148,\n",
       "       1.14594349e-150, 2.17885869e-153, 3.83872214e-156, 6.23965053e-159,\n",
       "       9.31077329e-162, 1.26806582e-164, 1.56551336e-167, 1.73772157e-170,\n",
       "       1.71711617e-173, 1.49200927e-176, 1.12229367e-179, 7.15976824e-183,\n",
       "       3.76671309e-186, 1.56897348e-189, 4.85149500e-193, 9.90000000e-197,\n",
       "       1.00000000e-200])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob_occurrences = stats.binom(n=100,p=0.01).pmf(number_events)\n",
    "prob_occurrences"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1bf2aafa-f1b5-48c6-98b3-323049a72c94",
   "metadata": {},
   "source": [
    "平均が（誤差を考えるとほぼ）1になることを確認します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "7cfbf663-acae-47fe-b010-024b9b55036d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9999999999999999"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(number_events * prob_occurrences)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5dafd5fe-4659-4414-952c-44ba8e9be831",
   "metadata": {},
   "source": [
    "分散を「２乗の平均　ー　平均の２乗」で計算すると、これも、ほぼ１になります。その理由はこのあと「ポワソン分布の確率関数」のところで説明します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "5b593b5b-2a53-48df-ab0c-3404adb9ebdf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.99"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(number_events * number_events *prob_occurrences) - (sum(number_events * prob_occurrences))**2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8efa921f-212c-42c9-b5de-5de66c75cce9",
   "metadata": {},
   "source": [
    "確率分布をプロットします。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "45aadcc6-31b7-47d8-a13c-63b39628971f",
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "0b0e30fb-14f9-415f-b8af-6b9c0b41a533",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x2d2edb9d340>]"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(number_events,prob_occurrences,'o',ms=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4112b2bb-20d3-42a1-912b-c5cae519fc20",
   "metadata": {},
   "source": [
    "このように、1回起こる確率が最も高く0.370なのですが、0回の確率はそれと同じくらい（0.367）で、2回起こっても不思議ではなく（0.185）、3回起こる確率も無視できません（0.0610）。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9d45f294-6071-4df7-8802-4289e4ec5e5f",
   "metadata": {},
   "source": [
    "### ポワソン分布の確率関数"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a0b38136-a2af-445b-8e48-0d321136f780",
   "metadata": {},
   "source": [
    "上の例のような二項分布は「ポワソン分布」によって近似されます。ポワソン分布は二項分布が次のような場合に生じます。  \n",
    "- 二項分布$B(n,p)$において期待値$np = \\lambda$を固定する。\n",
    "- 試行回数と成功確率について$n \\rightarrow \\infty$, $p \\rightarrow 0$のような極限をとる"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c3936b8b-bdbc-419e-815e-3fba519f090e",
   "metadata": {},
   "source": [
    "従って、ポワソン分布の確率関数は、$$f(x) = e^{-x}\\lambda^x / x! \\;\\;\\; (x = 0, 1, 2, ...)$$の形に表されます。パラメータは$\\lambda$だけになります。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c354f8b2-ec6f-43ae-86a1-4e508766fd2d",
   "metadata": {},
   "source": [
    "ポワソン分布の期待値$E[X]$と分散$V[X]$はいずれも$\\lambda$になります。  \n",
    "すなわち、$E[X] = V[X] = \\lambda$です。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "72fdc688-6827-44ff-b485-3068770b0dd5",
   "metadata": {},
   "source": [
    "### 例５ Scipyに含まれるポワソン分布の確率関数"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a00fbb32-8bf4-4f5a-b7af-25adf1cd5a6d",
   "metadata": {},
   "source": [
    "Scipyのstats.poisson.pmf([0,1,2,..,n],$\\lambda$)関数が、ポワソン分布において、$x = 0,1,2,...,n $に対する確率関数です。$\\lambda$は上で見たように、期待値（平均値）です。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5d34d343-c3a4-4b15-94e3-1ba6840de7ab",
   "metadata": {},
   "source": [
    "例４の計算を、「ポワソン分布」を仮定して行ってみましょう。SciPyのstats.poisson.pmf()関数を使います。  \n",
    "この関数の引数は、確率変数（この場合、0から100の配列）とパラメータmu（=平均）です。stats.binom.pmf()と同様に確率変数の配列を与えられたとき、それらのprobability mass functionを収納した配列を返します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "7f9019d8-b2da-4c7c-85d6-6353295eed1e",
   "metadata": {},
   "outputs": [],
   "source": [
    "prob_occurrences_2 = stats.poisson.pmf(number_events,1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "796a0f23-1905-487a-98ee-dbd0434deb22",
   "metadata": {},
   "source": [
    "二項分布で計算した確率分布とポワソン分布を仮定して計算した確率分布とに対して、プロットを重ね描きします。  \n",
    "すると、ほとんど重なっていることがわかります。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "cd10faff-65e1-4468-802a-c5d4623bf9fe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x2d2edd14760>]"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(number_events,prob_occurrences,'o',ms=1)\n",
    "plt.plot(number_events,prob_occurrences_2,'o',ms=1,color='r') #赤がポワソン分布"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "97059587-8e94-4958-9dd1-e476fe61945d",
   "metadata": {},
   "source": [
    "このように、この場合は、ポワソン分布で２項分布を代用できることがわかります。  \n",
    "ある単位（時間とか、面積とか）の中の期待値が1ぐらいのとき、ポワソン分布は２項分布の代用として使えます。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5ffc930b-7e33-4916-84c2-d9efd6dd13df",
   "metadata": {},
   "source": [
    "平均を計算します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "ed6e2c54-5784-4206-bdb6-651993e86b46",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.0\n"
     ]
    }
   ],
   "source": [
    "mu = sum(number_events * prob_occurrences_2)\n",
    "print(mu)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "08d3c2ec-1aa0-4932-98b5-fa37c534f932",
   "metadata": {},
   "source": [
    "分散を計算します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "d71e0735-6795-4c47-9129-40a226ba310b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum((number_events - mu)**2 * prob_occurrences_2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e8d2c745-9750-4fee-a211-dd2057282461",
   "metadata": {},
   "source": [
    "### パラメータ$\\lambda$の値によるポワソン分布の形の違い"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "83b925bb-4383-4857-bdd9-fc6adb43d748",
   "metadata": {},
   "source": [
    "さて、期待値が1のとき、ポワソン分布はこのような形になりましたが、期待値$\\lambda$が変わると、分布の形が変わります。  \n",
    "下のプログラムで、mu（$\\mu$）はポワソン分布の$\\lambda$と同じです。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "id": "42445949-3396-46c7-9b8b-77ee6fe442c1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x1af4f376040>"
      ]
     },
     "execution_count": 131,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "number_events_2 = np.arange(0,26,1)\n",
    "plt.plot(number_events_2,stats.poisson.pmf(number_events_2,1),'-o',ms=4, label='mu=1',color='b')\n",
    "plt.plot(number_events_2,stats.poisson.pmf(number_events_2,2),'-o',ms=4, label='mu=2',color='r')\n",
    "plt.plot(number_events_2,stats.poisson.pmf(number_events_2,3),'-o',ms=4, label='mu=3',color='y')\n",
    "plt.plot(number_events_2,stats.poisson.pmf(number_events_2,4),'-o',ms=4, label='mu=4',color='c')\n",
    "plt.plot(number_events_2,stats.poisson.pmf(number_events_2,5),'-o',ms=4, label='mu=5',color='m')\n",
    "plt.plot(number_events_2,stats.poisson.pmf(number_events_2,6),'-o',ms=4, label='mu=6',color='k')\n",
    "plt.plot(number_events_2,stats.poisson.pmf(number_events_2,7),'-o',ms=4, label='mu=7',color='g')\n",
    "\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "217bd8f6-109c-4118-8d02-f17db8bebde7",
   "metadata": {},
   "source": [
    "このように、単位時間・面積等における期待値が大きくなればなるほど、ポワソン分布は左右対称に近づきます。次に説明するように、正規分布に近づきます。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "678b1dc4-caec-473f-a403-53b9b1ccac57",
   "metadata": {},
   "source": [
    "### ポワソン分布の使い方"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "656422f2-bfc5-490a-a7f8-b6e5d29a765e",
   "metadata": {},
   "source": [
    "#### 練習問題１"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "417e6210-8d20-4530-beea-e194425d95f0",
   "metadata": {},
   "source": [
    "ある商品の需要が週平均３個のポワソン分布に従うものとする。  \n",
    "(a) 週間需要0の確率はどれほどか。  \n",
    "(b) 品切れの確率を5%に抑えるためには週初めの在庫をいくらにしておけばよいか。確率を1%に抑えるためにはどうするか。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ccdfd66a-58f8-4774-b150-f5ffecf11b61",
   "metadata": {},
   "source": [
    "解答：  \n",
    "(a) $P(0) = e^{-3} 3^0/0! = 0.050$ （計算は下欄ですがPythonを使うまでもなくGoogleでexp(-3)と検索すれば、できますね）"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "d1472cef-c810-46e0-a0f2-a1b3eea55c5b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.049787068367863944"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "np.exp(-3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "26151816-0ae6-4115-bbf4-9d94735bec5b",
   "metadata": {},
   "source": [
    "あるいは、stats.poisson(mu=3).pmf([0])とします。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "f62e7ce1-9121-4477-9601-91819f4c1c46",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.04978707])"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy import stats\n",
    "stats.poisson(mu=3).pmf([0])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bde11623-43cb-41fd-a96f-ca08b130a934",
   "metadata": {},
   "source": [
    "(b) 品切れ確率を5%に抑えるには、累積分布関数$F(X \\geqq x) \\leqq 0.05$となるような$x$を求めます。累積分布関数は通常は$F(X \\leqq x)$の形で定義され、ポワソン分布の場合、scipy.stats.poisson(mu=$\\mu$).ppf(q=$q$)を使います(PPF: Percent Point Function)。  \n",
    "この場合、需要がある個数を超える確率が0.05ということは、需要がある個数までに収まる累積確率が0.95ということですので、$q=0.95$となります。また、平均値$\\mu = 3$です。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "id": "511ed585-4f04-4be2-89cb-3c606474b2c8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.0"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.poisson(mu=3).ppf(q=0.95)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5ffc603c-eee8-439e-a53a-baabde49f942",
   "metadata": {},
   "source": [
    "品切れ確率を0.01に抑えるための在庫の計算には、同様にして、$F(X \\leqq x) = 0.99$となるような$x$を求めます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "id": "b10d1759-1129-4eee-bb9b-1faee000c416",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.0"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.poisson(mu=3).ppf(q=0.99)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "38c857ca-645e-4656-aa23-d820347e5e33",
   "metadata": {},
   "source": [
    "#### 練習問題２"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e0f46f30-3a7a-48fe-9ce5-2c026ae1d80c",
   "metadata": {},
   "source": [
    "ミサイルの部品の故障の確率はいずれも0.001で、ミサイルは500の部品でできているものとする。  \n",
    "(a) ミサイルを打ち上げて作動しない確率はどれほどか。  \n",
    "(b) 無事に作動する確率を0.90とするためには部品の信頼度をどの程度まで上げねばならないか。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0d69d2cb-4134-4215-9c95-b132f6d06889",
   "metadata": {},
   "source": [
    "解答のヒント：  \n",
    "(a) 故障部品数の期待値$np = 500 x 0.001 = 0.5$です。ポワソン分布を使って$F(X \\gt 0)$を求めましょう。  \n",
    "(b)故障部品数が0の確率$P(X=0)$を0.90以上にします。ポワソン分布の確率関数は  \n",
    "scipy.stats.poisson(mu=$\\mu$).pmf([$x_1, x_2, ...,x_n$])  \n",
    "ですが、ある確率変数に対する確率を与えられたときパラメータ$\\lambda = \\mu$を求めるための関数はないので、ポワソン分布の定義に従って計算します。  \n",
    "$P(X=x) = e^{-\\lambda}\\lambda^x/x!$ですから$P(X=0) = e^{-\\lambda}\\lambda^0/0! = e^{-\\lambda}$  \n",
    "です。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a3481c65-8b3c-4a67-a049-a931b89249f4",
   "metadata": {},
   "source": [
    "解答：  \n",
    "(a) $F(X \\gt 0) = 1 - F(X \\leqq 0) =  1 - P(X = 0) = 1 - e^{-0.5} = 1 - 0.607 = 0.393$　または、"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "id": "c8f397b0-2a07-4fa0-a0f5-518392fd93a2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.3934693402873666"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 - stats.poisson(mu=0.5).cdf(0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "11589518-57bb-4550-b34e-0c6ac1e91ccd",
   "metadata": {},
   "source": [
    "このミサイルはポンコツですね。そこで問(b)があります。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "76b6c235-b03c-484b-8c99-c0463b6d4b59",
   "metadata": {},
   "source": [
    "(b) $e^{-\\lambda} \\geqq 0.90$より、$\\lambda \\leqq ln(0.90) = -ln(0.90)$です  \n",
    "各部品の故障確率を$p$とすると$\\lambda = 500p$ですから、$p \\leqq -ln(0.90)/500 = 0.00021$  \n",
    "となるので、各部品の故障率は今の1/5のおよそ0.0002まで下げねばなりません。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a5e9d99f-fcb2-450f-b52c-0df4679abdef",
   "metadata": {},
   "source": [
    "## 正規分布"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7615d78d-3999-46f9-9d75-00bdd825bb30",
   "metadata": {},
   "source": [
    "ある期間に１回起こるぐらいのできごとについては、二項分布はポワソン分布で代用できることがわかりました。それでは、もっと回数が多くなると、どうでしょうか？  \n",
    "p=0.25で100回試行したときの確率分布を、二項分布とポワソン分布($\\mu$ =25（期待値が25）)とで重ね描きしてみます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "id": "f402ff59-d98d-425d-a126-9ac761c2487d",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import stats\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "id": "94da6947-2570-4cd5-83b6-9678a838f2f0",
   "metadata": {},
   "outputs": [],
   "source": [
    "number_events = np.arange(0,101,1) # 確率変数 X: x = 0, 1, 2, ..., 100\n",
    "prob_occurrences_3 = stats.binom(n=100,p=0.25).pmf(number_events) # 二項分布\n",
    "prob_occurrences_4 = stats.poisson.pmf(number_events,25) # ポワソン分布"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "74a61592-4545-488f-a71a-d8d651f87007",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x1af4a804640>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(number_events,prob_occurrences_3,'o',ms=1, label='binominal')\n",
    "plt.plot(number_events,prob_occurrences_4,'o',ms=1,color='r', label ='poisson') #赤がポワソン分布\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bdd92908-3199-4590-8fe6-f1b57c06dbf0",
   "metadata": {},
   "source": [
    "このように、二項分布とポワソン分布との分布の違いが顕著になります。  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7a3a2537-eecb-4094-9245-21ec3a26d8d8",
   "metadata": {},
   "source": [
    "このような場合、正規分布を使います。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8f80015f-d5f9-4453-ab8c-e54e8d3d59d3",
   "metadata": {},
   "source": [
    "### 正規分布の確率密度関数"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "52cc6aab-9468-4e44-9d26-088d8453220a",
   "metadata": {},
   "source": [
    "確率変数の配列$x$に対する正規分布の確率密度関数$f(x)$は、scipy.stats.norm.pdf(x,loc=$\\mu$,scale=$\\sigma$)で得られます。ただし、$\\mu$は平均、$\\sigma^2$は分散（つまり$\\sigma$は標準偏差）です。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1791c811-70b6-4d10-ab2f-7ee0494ce93b",
   "metadata": {},
   "source": [
    "この場合、$x = [0,1,2,...,100]$つまり0から100までの整数の配列です（下のnumber_events）また、平均$\\mu$=100xp = 100x0.25 = 25です。標準偏差$\\sigma$は、二項分布の分散すなわち$100*0.25*(1-0.25)$の平方根です。従って、確率分布は次のようになります。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "id": "54f5e3f2-bd11-4089-9d13-b17cd00a5a38",
   "metadata": {},
   "outputs": [],
   "source": [
    "prob_occurrences_5 = stats.norm.pdf(number_events,loc=25,scale=np.sqrt(100*0.25*0.75))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "141a17d8-a102-4963-89c6-bc00670a5835",
   "metadata": {},
   "source": [
    "この確率分布の、平均値周辺の値を見てみましょう。平均値25に対する確率密度関数の値は、配列の添字25に入っています。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "cdc82876-64f2-408c-b010-a958c83f7663",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.08281036, 0.08970739, 0.09213177, 0.08970739, 0.08281036])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob_occurrences_5[23:28]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cab109a3-b67e-49fc-a910-1b46291bd30f",
   "metadata": {},
   "source": [
    "同じ範囲の確率変数に対する二項分布の確率とくらべてみます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "382e76db-4186-4dab-8351-ddd4d51c574c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.08470922, 0.0905918 , 0.09179969, 0.08826893, 0.08064075])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prob_occurrences_3[23:28]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "5e08b1f4-3cf3-455c-a056-bef88d590086",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x1af4bd0ffa0>"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(number_events,prob_occurrences_3,'o',ms=1, label='binominal')\n",
    "plt.plot(number_events,prob_occurrences_5,'o',ms=1,color='r', label ='normal') #赤が正規分布\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2c17366e-a094-4f09-ab86-e4d4a8a9fd43",
   "metadata": {},
   "source": [
    "このように、ほとんど一致しました。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f3a8206a-b5e9-4286-8b4a-b2fd21d18573",
   "metadata": {},
   "source": [
    "なお、数学的には、$$f(x) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}}e^{-\\frac{(x - \\sigma)^2}{2\\sigma^2}}$$となります。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8d6600ac-a48f-4f5f-a079-3c7380ffcaa6",
   "metadata": {},
   "source": [
    "### 正規分布を合併したものの性質"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c8f4d438-7bba-43dd-933d-53304d5a5208",
   "metadata": {},
   "source": [
    "正規分布には次のような性質があります。\n",
    "- 確率変数が、平均$\\mu$, 分散$\\sigma^2$の正規分布$N(\\mu, \\sigma^2)$に従うとき、Xの１次関数$aX + b$も正規分布に従い、その平均は$E(aX + b) = a\\mu + b$、分散は$V(aX + b) = a^2\\sigma^2$となります。すなわち、$aX + b$は正規分布$N(a\\mu + b, a^2\\sigma^2)$に従います。\n",
    "- 上の特殊な場合として、$Z = \\frac{X - \\mu}{\\sigma}$と変換すると、$Z$は平均0, 分散1の正規分布$N(0,1)$（標準正規分布）に従います。このとき、$$f(x)=\\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{x^2}{2}}$$となります。\n",
    "- 確率変数$X$と$Y$とが独立に正規分布$N(\\mu_1, \\sigma_1^2)$, $N(\\mu_2, \\sigma_2^2)$に従うとき、$X + Y$も正規分布に従い、その分布は$N(\\mu_1 + \\mu_2, \\sigma_1^2 + \\sigma_2^2)$になります。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ca7ec0e-ce35-4460-aa40-790b70e86528",
   "metadata": {},
   "source": [
    "#### 実験1 正規分布の合併"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bb2c1731-5ceb-4b8b-8c0c-5b0391278d72",
   "metadata": {},
   "source": [
    "このように、正規分布の性質から、正規分布を合併したものもまた正規分布になります。シミュレーションで確かめてみましょう。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c80574f2-8cfc-486c-80a5-82f7001ff880",
   "metadata": {},
   "source": [
    "- 集団A: 平均75, 分散25（標準偏差5）の正規分布を示す1000個の数  \n",
    "- 集団B: 平均30, 分散16（標準偏差4）の正規分布を示す1000個の数  \n",
    "を次のようにして生成します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "id": "a50987e7-970f-4c94-a626-151777629131",
   "metadata": {},
   "outputs": [],
   "source": [
    "generator = np.random.default_rng(12345)\n",
    "pop_A = generator.normal(loc=75, scale=5, size=1000)\n",
    "pop_B = generator.normal(loc=30, scale=4, size=1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b4ad8bd8-c439-4de7-af08-a08ff6d7cd0f",
   "metadata": {},
   "source": [
    "生成されたそれぞれの集合の平均と分散を調べましょう。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "id": "5e481b14-9b78-45d3-82d4-42edd70f008b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "75.04129196042138\n",
      "25.767588547481136\n",
      "30.145524791021423\n",
      "16.226486888762388\n"
     ]
    }
   ],
   "source": [
    "print(np.mean(pop_A))\n",
    "print(np.var(pop_A))\n",
    "print(np.mean(pop_B))\n",
    "print(np.var(pop_B))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bf873f96-7a41-4149-a60d-fdcf847a9fa3",
   "metadata": {},
   "source": [
    "集団Aのヒストグラムは次のようになります。なお、階級の端の値を決めるために最大値と最小値を調べました。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "id": "dbda4ada-5b6f-40b0-b57b-788b3f3d377e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([  0.,   2.,  10.,  15.,  35.,  54.,  98., 119., 163., 152., 136.,\n",
       "         93.,  66.,  35.,  14.,   5.,   1.,   2.,   0.]),\n",
       " array([57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89,\n",
       "        91, 93, 95]),\n",
       " <BarContainer object of 19 artists>)"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "intervals_a = np.arange(57,97,2) # ヒストグラムの階級の左端の値の配列。中心が58, 60, 62, ...94になるように。\n",
    "plt.hist(pop_A,bins=intervals_a) # 集団Aの確率変数のヒストグラムを描く。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "88639498-97aa-4230-88bf-c8ffa3a9b2ff",
   "metadata": {},
   "source": [
    "集団Bのヒストグラムは次のようになります。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "id": "94a4c15d-d353-4831-aa22-328310d04c66",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([  0.,   1.,  17.,  22.,  62.,  99., 179., 212., 173., 126.,  63.,\n",
       "         32.,   7.,   5.,   2.]),\n",
       " array([15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45]),\n",
       " <BarContainer object of 15 artists>)"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAAD4CAYAAAAXUaZHAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMCwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8/fFQqAAAACXBIWXMAAAsTAAALEwEAmpwYAAARJUlEQVR4nO3dfYxldX3H8fdHRNtoDeBONxvADtLVhpq62CmloRoqtV3BCDSGQiyiJV1IIMFoqgtNim1qsrYirWmLWQplTZGHigQitHWDRGJS0AG2uDyVB5ewm2V3fMZqSIFv/5gz6XWY2b0z987O3B/vV3Iz5/zO0/eXs/vZs785555UFZKktrxiuQuQJA2f4S5JDTLcJalBhrskNchwl6QGvXK5CwBYtWpVjY+PL3cZkjRS7r333u9U1dhcy1ZEuI+PjzM5ObncZUjSSEny1HzLHJaRpAYZ7pLUIMNdkhpkuEtSgwx3SWqQ4S5JDTLcJalBhrskNchwl6QGrYgnVKXlML7xtqHub8emU4a6P2kQXrlLUoMMd0lqkOEuSQ0y3CWpQYa7JDXIcJekBhnuktSg/YZ7kiOT3JnkoSQPJrmoaz8sydYkj3U/D+3ak+SzSR5P8kCSty11JyRJP6ufK/fngY9W1THA8cAFSY4BNgJ3VNVa4I5uHuDdwNruswG4YuhVS5L2ab/hXlW7q+q+bvpZ4GHgcOBUYEu32hbgtG76VODzNe1u4JAka4ZduCRpfgsac08yDhwL3AOsrqrd3aJngNXd9OHA0z2b7ezaZu9rQ5LJJJNTU1MLrVuStA99h3uS1wI3AR+uqh/1LquqAmohB66qzVU1UVUTY2NjC9lUkrQffYV7koOZDvZrq+pLXfOemeGW7ufern0XcGTP5kd0bZKkA6Sfu2UCXAU8XFWf6Vl0K3BON30OcEtP+we6u2aOB37YM3wjSToA+vnK3xOAs4FvJdnWtV0CbAJuTHIu8BRwRrfsduBk4HHgJ8CHhlmwJGn/9hvuVfV1IPMsPmmO9Qu4YMC6JEkD8AlVSWqQ4S5JDTLcJalBhrskNchwl6QGGe6S1CDDXZIaZLhLUoMMd0lqkOEuSQ0y3CWpQYa7JDXIcJekBhnuktQgw12SGtTPyzqkZTe+8bblLkEaKf28Zu/qJHuTbO9puyHJtu6zY+YNTUnGk/y0Z9nnlrB2SdI8+rlyvwb4e+DzMw1V9Ycz00kuA37Ys/4TVbVuSPVJkhahn9fs3ZVkfK5l3cuzzwDeOeS6JEkDGPQXqm8H9lTVYz1tRyW5P8nXkrx9vg2TbEgymWRyampqwDIkSb0GDfezgOt65ncDb6iqY4GPAF9I8rq5NqyqzVU1UVUTY2NjA5YhSeq16LtlkrwS+APg12faquo54Llu+t4kTwBvAiYHrFNa8YZ9R8+OTacMdX96eRnkyv13gUeqaudMQ5KxJAd1028E1gJPDlaiJGmh+rkV8jrgP4E3J9mZ5Nxu0Zn87JAMwDuAB7pbI78InF9V3xtivZKkPvRzt8xZ87R/cI62m4CbBi9LkjQIv35AkhpkuEtSgwx3SWqQ4S5JDTLcJalBhrskNchwl6QGGe6S1CDDXZIaZLhLUoMMd0lqkOEuSQ0y3CWpQYa7JDXIcJekBvXzso6rk+xNsr2n7RNJdiXZ1n1O7ll2cZLHkzya5PeXqnBJ0vz6uXK/Blg/R/vlVbWu+9wOkOQYpt/Q9KvdNv8489o9SdKBs99wr6q7gH5flXcqcH1VPVdV3wYeB44boD5J0iIMMuZ+YZIHumGbQ7u2w4Gne9bZ2bW9RJINSSaTTE5NTQ1QhiRptsWG+xXA0cA6YDdw2UJ3UFWbq2qiqibGxsYWWYYkaS6LCveq2lNVL1TVi8CV/P/Qyy7gyJ5Vj+jaJEkH0KLCPcmantnTgZk7aW4Fzkzy6iRHAWuBbwxWoiRpoV65vxWSXAecCKxKshO4FDgxyTqggB3AeQBV9WCSG4GHgOeBC6rqhSWpXJI0r/2Ge1WdNUfzVftY/5PAJwcpSpI0GJ9QlaQGGe6S1CDDXZIaZLhLUoMMd0lqkOEuSQ0y3CWpQYa7JDXIcJekBhnuktQgw12SGmS4S1KDDHdJapDhLkkNMtwlqUGGuyQ1aL/hnuTqJHuTbO9p+5skjyR5IMnNSQ7p2seT/DTJtu7zuSWsXZI0j36u3K8B1s9q2wq8pap+Dfhv4OKeZU9U1bruc/5wypQkLUQ/r9m7K8n4rLav9MzeDbxvyHVJL3vjG28b6v52bDplqPvTyjaMMfc/Bv6tZ/6oJPcn+VqSt8+3UZINSSaTTE5NTQ2hDEnSjIHCPcmfAc8D13ZNu4E3VNWxwEeALyR53VzbVtXmqpqoqomxsbFBypAkzbLocE/yQeA9wPurqgCq6rmq+m43fS/wBPCmIdQpSVqARYV7kvXAx4D3VtVPetrHkhzUTb8RWAs8OYxCJUn92+8vVJNcB5wIrEqyE7iU6btjXg1sTQJwd3dnzDuAv0zyv8CLwPlV9b0lql2SNI9+7pY5a47mq+ZZ9ybgpkGLkiQNxidUJalBhrskNchwl6QGGe6S1CDDXZIaZLhLUoMMd0lqkOEuSQ0y3CWpQYa7JDXIcJekBhnuktQgw12SGmS4S1KDDHdJalBf4Z7k6iR7k2zvaTssydYkj3U/D+3ak+SzSR5P8kCSty1V8ZKkufV75X4NsH5W20bgjqpaC9zRzQO8m+nX660FNgBXDF6mJGkh+gr3qroLmP26vFOBLd30FuC0nvbP17S7gUOSrBlCrZKkPg0y5r66qnZ3088Aq7vpw4Gne9bb2bX9jCQbkkwmmZyamhqgDEnSbPt9h2o/qqqS1AK32QxsBpiYmFjQtlr5xjfettwlSC9rg1y575kZbul+7u3adwFH9qx3RNcmSTpABgn3W4FzuulzgFt62j/Q3TVzPPDDnuEbSdIB0NewTJLrgBOBVUl2ApcCm4Abk5wLPAWc0a1+O3Ay8DjwE+BDQ65ZkrQffYV7VZ01z6KT5li3gAsGKUqSNBifUJWkBhnuktQgw12SGmS4S1KDDHdJapDhLkkNMtwlqUGGuyQ1yHCXpAYZ7pLUIMNdkhpkuEtSgwx3SWqQ4S5JDTLcJalBi36HapI3Azf0NL0R+HPgEOBPgJm3Xl9SVbcv9jiSpIVbdLhX1aPAOoAkBzH9ntSbmX7z0uVV9elhFChJWrhhDcucBDxRVU8NaX+SpAEMK9zPBK7rmb8wyQNJrk5y6JCOIUnq08DhnuRVwHuBf+2argCOZnrIZjdw2TzbbUgymWRyampqrlUkSYs0jCv3dwP3VdUegKraU1UvVNWLwJXAcXNtVFWbq2qiqibGxsaGUIYkacYwwv0seoZkkqzpWXY6sH0Ix5AkLcCi75YBSPIa4F3AeT3Nf51kHVDAjlnLJEkHwEDhXlX/A7x+VtvZA1UkSRqYT6hKUoMMd0lqkOEuSQ0y3CWpQYa7JDXIcJekBg10K6Sk0TG+8bah73PHplOGvk8Nh+EuYGn+4ktaPg7LSFKDDHdJapDhLkkNMtwlqUGGuyQ1yHCXpAYZ7pLUIMNdkho08ENMSXYAzwIvAM9X1USSw4AbgHGm38Z0RlV9f9BjSZL6M6wr99+pqnVVNdHNbwTuqKq1wB3dvCTpAFmqYZlTgS3d9BbgtCU6jiRpDsMI9wK+kuTeJBu6ttVVtbubfgZYPXujJBuSTCaZnJqaGkIZkqQZw/jisN+uql1JfhHYmuSR3oVVVUlq9kZVtRnYDDAxMfGS5ZKkxRv4yr2qdnU/9wI3A8cBe5KsAeh+7h30OJKk/g0U7klek+QXZqaB3wO2A7cC53SrnQPcMshxJEkLM+iwzGrg5iQz+/pCVf17km8CNyY5F3gKOGPA40iSFmCgcK+qJ4G3ztH+XeCkQfYtSVo8n1CVpAYZ7pLUIMNdkhpkuEtSgwx3SWqQ4S5JDTLcJalBhrskNchwl6QGGe6S1CDDXZIaZLhLUoOG8bIOSS9T4xtvG+r+dmw6Zaj7eznzyl2SGmS4S1KDFh3uSY5McmeSh5I8mOSirv0TSXYl2dZ9Th5euZKkfgwy5v488NGquq971d69SbZ2yy6vqk8PXp4kaTEWHe5VtRvY3U0/m+Rh4PBhFSZJWryhjLknGQeOBe7pmi5M8kCSq5McOs82G5JMJpmcmpoaRhmSpM7A4Z7ktcBNwIer6kfAFcDRwDqmr+wvm2u7qtpcVRNVNTE2NjZoGZKkHgOFe5KDmQ72a6vqSwBVtaeqXqiqF4ErgeMGL1OStBCLHnNPEuAq4OGq+kxP+5puPB7gdGD7YCVqtmE/OCKpPYPcLXMCcDbwrSTburZLgLOSrAMK2AGcN8AxJEmLMMjdMl8HMsei2xdfjiRpGHxCVZIaZLhLUoP8VsgDwF+ASjrQvHKXpAYZ7pLUIMNdkhpkuEtSgwx3SWqQ4S5JDTLcJalBhrskNchwl6QG+YSqpBVj2E9z79h0ylD3N0q8cpekBnnlLqlZS/G9TqPyvwGv3CWpQUsW7knWJ3k0yeNJNi7VcSRJL7UkwzJJDgL+AXgXsBP4ZpJbq+qhpTieJB0oo/JL36Uacz8OeLyqngRIcj1wKjAS4e73r0sadUsV7ocDT/fM7wR+s3eFJBuADd3sj5M8OsDxVgHfGWD7laKVfoB9WYla6Qc01Jd8aqC+/NJ8C5btbpmq2gxsHsa+kkxW1cQw9rWcWukH2JeVqJV+gH3px1L9QnUXcGTP/BFdmyTpAFiqcP8msDbJUUleBZwJ3LpEx5IkzbIkwzJV9XySC4H/AA4Crq6qB5fiWJ2hDO+sAK30A+zLStRKP8C+7Feqain2K0laRj6hKkkNMtwlqUEjFe5Jrk6yN8n2nrZPJNmVZFv3OXk5a+xXkiOT3JnkoSQPJrmoaz8sydYkj3U/D13uWvdlH/0YufOS5OeSfCPJf3V9+Yuu/agk93RfpXFDd5PAiraPvlyT5Ns952XdMpfalyQHJbk/yZe7+ZE7JzPm6MuSnJORCnfgGmD9HO2XV9W67nP7Aa5psZ4HPlpVxwDHAxckOQbYCNxRVWuBO7r5lWy+fsDonZfngHdW1VuBdcD6JMcDn2K6L78MfB84d/lK7Nt8fQH4057zsm25Clygi4CHe+ZH8ZzMmN0XWIJzMlLhXlV3Ad9b7jqGoap2V9V93fSzTJ/sw5n+moYt3WpbgNOWpcA+7aMfI6em/bibPbj7FPBO4Itd+4o/J7DPvoycJEcApwD/1M2HETwn8NK+LKWRCvd9uDDJA92wzYoexphLknHgWOAeYHVV7e4WPQOsXq66FmpWP2AEz0v3X+ZtwF5gK/AE8IOqer5bZScj8o/X7L5U1cx5+WR3Xi5P8urlq7Bvfwt8DHixm389I3pOeGlfZgz9nLQQ7lcARzP9X8/dwGXLWs0CJXktcBPw4ar6Ue+ymr5PdSSutubox0iel6p6oarWMf1U9XHAryxvRYs3uy9J3gJczHSffgM4DPj48lW4f0neA+ytqnuXu5ZB7aMvS3JORj7cq2pP94f4ReBKpv9CjoQkBzMdiNdW1Ze65j1J1nTL1zB91bWizdWPUT4vAFX1A+BO4LeAQ5LMPPA3cl+l0dOX9d0wWlXVc8A/s/LPywnAe5PsAK5nejjm7xjNc/KSviT5l6U6JyMf7jNB2Dkd2D7fuitJN254FfBwVX2mZ9GtwDnd9DnALQe6toWYrx+jeF6SjCU5pJv+eabfR/Aw08H4vm61FX9OYN6+PNJz4RCmx6lX9Hmpqour6oiqGmf6a0y+WlXvZwTPyTx9+aOlOicj9Q7VJNcBJwKrkuwELgVO7G4dKmAHcN5y1bdAJwBnA9/qxkUBLgE2ATcmORd4Cjhjecrr23z9OGsEz8saYEumXzbzCuDGqvpykoeA65P8FXA/0/+YrXTz9eWrScaAANuA85exxkF8nNE7J/O5dinOiV8/IEkNGvlhGUnSSxnuktQgw12SGmS4S1KDDHdJapDhLkkNMtwlqUH/B6BKJvs+QDNtAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "intervals_b = np.arange(15,47,2) # ヒストグラムの階級の左端の値の配列。中心が16, 18, 20, ...46になるように。\n",
    "plt.hist(pop_B,bins=intervals_b) # 集団Aの確率変数のヒストグラムを描く。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c5b50a06-d04a-48f9-b02b-c4a3e52020d8",
   "metadata": {},
   "source": [
    "このとき、集団C：集団Aと集団Bの値を足したもの　とします。  \n",
    "「値を足す」とは、対応する要素の値を足すことで、例えば、タプル(1,2,3)と(4,5,6)からは(5,7,9)ができます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "id": "897de0d4-88e1-4440-b3bb-d123e1d9514b",
   "metadata": {},
   "outputs": [],
   "source": [
    "pop_C = pop_A + pop_B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "id": "4bee56ba-c1d3-4852-bc46-1397480ddc9d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "105.18681675144282\n",
      "42.94173938544988\n",
      "105.18681675144279\n",
      "41.99407543624352\n"
     ]
    }
   ],
   "source": [
    "print(np.mean(pop_C))\n",
    "print(np.var(pop_C))\n",
    "print(np.mean(pop_A)+np.mean(pop_B))\n",
    "print(np.var(pop_A)+np.var(pop_B))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "448f9dae-4fec-4f8a-9788-7cff91ae9ea6",
   "metadata": {},
   "source": [
    "となり、集団Cの平均ならびに分散が、それぞれ、集団AとBとの平均の和、ならびに、分散の和にほぼ等しいことがわかります。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1e0c02d4-cd67-434c-8f2e-50ed95b11c7d",
   "metadata": {},
   "source": [
    "そこで、集団Cの、理論上の平均（105）と分散（41）に等しいような、正規分布$N(105,41)$を作り、重ねてみましょう。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "id": "80837baa-084d-40c3-b006-2a60c9e73c55",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x2305d61b790>"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# figureインスタンスを作る\n",
    "fig = plt.figure()\n",
    "# Y軸を2つ作る。\n",
    "ax1 = fig.subplots() # 1つめのY軸（左側）\n",
    "ax2 = ax1.twinx() # 2つめのY軸　（右側）\n",
    "\n",
    "x_val = np.arange(80,131,0.1) # 正規分布の確率分布をプロットするための、確率変数の配列\n",
    "intervals = np.arange(79,133,2) # ヒストグラムの階級の左端の値の配列。中心が80, 82, 84, ...130になるように。\n",
    "\n",
    "ax1.hist(pop_C,bins=intervals) # 集団Ｃの確率変数のヒストグラムを描く。Y軸は左側。\n",
    "# 次に正規分布の確率分布をプロットする。Y軸は右側\n",
    "#集団Cの平均と標準偏差から仮定される正規分布\n",
    "prob_occurrences_7 = stats.norm.pdf(x_val,loc=105,scale=np.sqrt(41))\n",
    "ax2.plot(x_val,prob_occurrences_7,'o',ms=1,color='b', label='Probability')\n",
    "# 次に、正規分布の確率分布に従ったとき、ヒストグラムの各階級の中心の値に対して頻度を予測する。\n",
    "points_1 = np.arange(80,132,2) # 予測対象のX軸の値の配列\n",
    "ax1.set_ylabel(\"Data points\", fontsize=10) # 左側のY軸のラベル\n",
    "ax2.set_ylabel(\"Probability\", fontsize=10) # 右側のY軸のラベル\n",
    "\n",
    "# 両方のY軸の凡例を重ならないように描くための方法\n",
    "h1, l1 = ax2.get_legend_handles_labels()\n",
    "h2, l2 = ax1.get_legend_handles_labels()\n",
    "ax1.legend(h1 + h2, l1 + l2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c37e2e4b-d03e-4d62-a890-9b2263de7a06",
   "metadata": {},
   "source": [
    "### 正規分布の使い方"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "71b50c6f-26ef-4eb8-9f96-c90629b3eae4",
   "metadata": {},
   "source": [
    "正規分布は連続関数つまり「確率密度関数」です。ですから、確率変数が区間[a, b]をとる確率がどれだけか、という使い方をします。また、標準正規分布になおして使うことも多いです。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7bfe9355-aaee-428e-bd8c-2bb1ae91ef80",
   "metadata": {},
   "source": [
    "#### 練習問題３ 偏差値と割合"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8181f941-90b7-4bf3-9d50-a1635b6b92ee",
   "metadata": {},
   "source": [
    "偏差値  $\\equiv 10 \\times \\frac{X - \\mu}{\\sigma} + 50 = 10Z + 50$です。得点が正規分布だとして、偏差値60以上の人の割合はどれだけですか？"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f4ebd700-986f-4b47-a6ff-34ce133755f4",
   "metadata": {},
   "source": [
    "ヒント：  \n",
    "- $Z \\geqq 1$です。\n",
    "- $Z$が1以下の人の割合は、標準正規分布$N(0,1)$に対して累積分布関数$F(X \\leqq 1)$を求めます。\n",
    "- 標準正規分布$N(0,1)$の$F(X \\leqq 1)$は、scipy.stats.norm(loc=0,scale=1).cdf(1)で求められます。\n",
    "- $F(X \\geqq 1)$は、1から上の値を引くか、scipy.stats.norm(loc=0,scale=1).sf(1)で求められます。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e3f5d38e-fc44-4f15-85b2-e842754fec95",
   "metadata": {},
   "source": [
    "解答："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "id": "5c91b19f-b412-4f8c-9cb7-c655f034d644",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.15865525393145707"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy import stats\n",
    "stats.norm(loc=0,scale=1).sf(1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fd0ad2a9-4a17-4a71-97ee-ca2d31e23207",
   "metadata": {},
   "source": [
    "#### 練習問題４ 連続的な値で品質が表される場合の、不良品の確率"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "308ad7e9-b747-4150-8efd-0b82a0938b70",
   "metadata": {},
   "source": [
    "ピーナツの缶に正味量1ポンドと表示されている。缶詰め作業は$\\mu=1.05$ポンド、$\\sigma = 0.02$ポンドで工程管理されている。中身が表示量に達しない確率を求めよ。缶の内容量$x$は正規分布に従うものとする。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "64a19928-2268-49b7-bf4b-2025ea3a07de",
   "metadata": {},
   "source": [
    "ヒント：中身が1ポンドのときの$Z = \\frac{1 - 1.05}{0.02} = -2.5$です。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fea87795-88c9-4960-8254-eb87d4fab2a9",
   "metadata": {},
   "source": [
    "解答："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "id": "05ddfb1f-2da8-4abf-9923-d2b50b07d9fe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.006209665325776132"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy import stats\n",
    "stats.norm(loc=0,scale=1).cdf(-2.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6c6f585b-04eb-482a-91ed-5737830ec45d",
   "metadata": {},
   "source": [
    "#### 練習問題５ 求められる品質管理の程度"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "83f7d73d-0330-4679-84b4-d6188cb89291",
   "metadata": {},
   "source": [
    "ある製品の寿命が平均200時間だとして、この製品の90%以上が150時間以上の寿命を有するためには、寿命分布の標準偏差を最大どれだけに抑えることが必要か？"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3cb83bcf-7861-452b-ab3f-beccc5be7579",
   "metadata": {},
   "source": [
    "ヒント：  \n",
    "- 寿命が150時間に満たない製品の割合は、0.1です\n",
    "- まず、パーセント・ポイント関数 scipy.stats.norm.ppf()を使って、$Z$を求めます。\n",
    "- $Z = \\frac{X - \\mu}{\\sigma}$より、$\\sigma = \\frac{X - \\mu}{Z}$です。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1776330a-50ea-4fc1-8305-9afe14be2300",
   "metadata": {},
   "source": [
    "解答："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "id": "0601def1-4e95-4b78-8f19-562fdc41fa24",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-1.2815515655446004\n",
      "39.0 時間\n"
     ]
    }
   ],
   "source": [
    "z_min = stats.norm(loc=0,scale=1).ppf(0.1)\n",
    "print(z_min)\n",
    "sigma_min = (150 - 200)/z_min\n",
    "print(round(sigma_min,1),\"時間\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4a7a3124-7a74-4baf-9297-12eb93ec2144",
   "metadata": {},
   "source": [
    "## 指数分布"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "11d9f6ce-487b-4fc0-bde3-476ba21c6198",
   "metadata": {},
   "source": [
    "### 指数分布の定義"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fe186a89-42c9-46f5-a7ef-eaab5a7204a2",
   "metadata": {},
   "source": [
    "ポワソン分布に従うような事象が起こる間隔は、「指数分布」に従います。  \n",
    "すなわち、ポワソン分布の定数$\\lambda$（すなわち、単位時間や単位面積等における平均$\\mu$に相当）に対して、その確率密度関数は  \n",
    "$$\n",
    "    % コメントは\"%\"で始める\n",
    "    % 空白は表示に影響しないので、連続する記号と記号とは空白で区別する\n",
    "    % （空白を入れたければ\"\\quad\"を使う\n",
    "    % 下付き文字は\"_a\"、上付き文字は\"^a\"\n",
    "    % 改行は\"\\\\\"を付ける\n",
    "    f_{(x)} = \\lambda e^{-\\lambda x}\\\\\n",
    "    % 複数文字を1要素とする際は{...}で囲う\n",
    "$$\n",
    "となります。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ad427896-0fbd-4949-9ada-b9f876f023e5",
   "metadata": {},
   "source": [
    "### 指数分布の期待値と分散"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7590086b-c3d4-45bf-adf0-9e94f45595f4",
   "metadata": {},
   "source": [
    "ポワソン分布では、期待値も分散も$\\lambda$になりますが、指数分布では、期待値も標準偏差も$\\frac{1}{\\lambda}$となります。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d60c9cfb-ec9a-4205-a28c-85b55fbe1525",
   "metadata": {},
   "source": [
    "### 例：scipy.statsでの指数分布の扱い"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4a9fc823-0d23-438d-9b1a-83a3ecfe729b",
   "metadata": {},
   "source": [
    "さきほどの100年に１回の事象がおこる間隔の確率分布は、対応するポワソン分布が、1単位（100年間）に平均1, 分散1（つまり標準偏差1）ですので、stats.expon.pdf()関数を用いて、次のように得られます。横軸の単位は、100年になります。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "id": "7bac6ba2-4f35-4d21-a743-5ccb3f394f79",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import stats\n",
    "intervals_2 = np.arange(0,11,0.1)\n",
    "prob_occurrences_6 = stats.expon.pdf(intervals_2,scale=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "id": "564fa4f2-9564-44d8-869b-5c8cd453586a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x2305eb019d0>]"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(intervals_2,prob_occurrences_6,'o',ms=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "95f880ef-b95b-49ee-bb3d-5efdd370eb9e",
   "metadata": {},
   "source": [
    "#### 注意：指数分布の確率密度関数の意味とその利用法"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5b11fa24-ba12-43e2-90b7-4ffe6ebcab89",
   "metadata": {},
   "source": [
    "指数分布の確率密度関数は、指数分布の定義より、事象が「ｘ単位経過後の時間までは起こらず、ｘ単位経過後で始めておこる」確率を示します。事象どうしは独立なので、今起こる確率と今の直後に起こる確率は同じです。そして、時間が経つほどその確率は小さくなります。  \n",
    "ですから、より実用的な、「今からｘ単位後までに起こる」確率の計算のためには、これを時間0からxまで積分して累積確率密度を求める必要があります。stats.expon.cdf()関数を用います。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "c4f0cb04-7949-4901-9083-03d6155404eb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1801109f220>]"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "prob_occurrences_7 = stats.expon.cdf(intervals_2,scale=1) # 累積確率密度分布\n",
    "plt.plot(intervals_2,prob_occurrences_7,'o',ms=3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3a7024e2-0e1c-4bc3-935c-5ad46fa5b001",
   "metadata": {},
   "source": [
    "ちょうど1単位(100年)のところで確率が0.5になっているのがわかります。つまり、ここまではまだ起こらない確率のほうが高く、これ以降は、すでに起こっている確率のほうが高くなります。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "361def68-deb7-491c-8a2e-2d639f20e282",
   "metadata": {},
   "source": [
    "### 指数分布を使う"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "756946cc-01e4-401d-b41c-4b02bedefd6e",
   "metadata": {},
   "source": [
    "#### 練習問題6 イベントの間隔"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4b62d9dc-f1dc-4b77-a69b-329ce7cae89e",
   "metadata": {},
   "source": [
    "あるコールセンターに電話が１時間平均240回の割合でかかっている。次の電話が20秒以内にかかる確率はどれほどか。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9aa5560f-ed10-4b14-9f1f-a5678a433e43",
   "metadata": {},
   "source": [
    "ヒント：\n",
    "- 電話の平均到着間隔は、60x60/240 = 15[秒]です。\n",
    "- 平均15の指数分布に対して、$F(X \\leqq 20)$は、stats.expon(scale=15).cdf(20)で求めます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "id": "aadd2007-7647-40ee-84db-a396c896a650",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7364028618842733"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy import stats\n",
    "stats.expon(scale=15).cdf(20)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9a0e29da-8612-42f5-aff0-21c463f57c5c",
   "metadata": {},
   "source": [
    "#### 練習問題７　ポワソン分布のパラメータを与えられた場合"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "787ef50d-f201-40fc-b950-ced9ce4271da",
   "metadata": {},
   "source": [
    "ある売り場への客の到着数が１分間に平均４人のポワソン分布に従うものとする。  \n",
    "(a) 到着間隔が2分を超える確率を求めよ  \n",
    "(b) 到着間隔が30秒以下の確率を求めよ"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "619d40f3-4384-42af-a1c0-bfcf68c7973e",
   "metadata": {},
   "source": [
    "ヒント：  \n",
    "- ポワソン分布のパラメータ$\\lambda$に対し、平均間隔は$\\frac{1}{\\lambda}$単位時間になります。\n",
    "- (a) では、scipy.stats.expon.sf()関数を用います。scaleパラメータは平均間隔です。\n",
    "- (b) では、scipy.stats.expon.cdf()関数を用います。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fb2ee311-9de7-4382-9683-7adbb2ec6299",
   "metadata": {},
   "source": [
    "解答：(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "id": "99879839-ebca-4f66-878e-e431e488ff04",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.00033546262790251185"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy import stats\n",
    "stats.expon(scale=0.25).sf(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ba03d6cc-5907-42ee-b183-ce405ac09764",
   "metadata": {},
   "source": [
    "解答：(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "id": "89f5071c-675e-49a2-8f2e-d4cd9f5320a6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8646647167633873"
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.expon(scale=0.25).cdf(0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "40693e2a-5bf6-4430-98bd-df749c5b21d5",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
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   "language": "python",
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  "language_info": {
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