{ "nbformat": 4, "nbformat_minor": 0, "metadata": { "colab": { "provenance": [], "toc_visible": true }, "kernelspec": { "name": "ir", "display_name": "R" }, "language_info": { "name": "R" } }, "cells": [ { "cell_type": "markdown", "source": [ "# 第5回 標本分布" ], "metadata": { "id": "obnzB1V21pax" } }, { "cell_type": "markdown", "source": [ "[![open in colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/slt666666/biostatistics_text_wed/blob/main/source/_static/colab_notebook/chapter5.ipynb)" ], "metadata": { "id": "O7LNMBgAby9C" } }, { "cell_type": "markdown", "source": [ "## はじめに\n", "\n", "統計学では、ある集団の特徴を知りたいときに、集団全体のデータからではなく、一部のデータから特徴を推測することが多いです。\n", "\n", "例えば「京都府の中に、京都大学を卒業した人がどのくらいの割合いるのか」を知りたい時に、京都府民全員の学歴を調査するのは途方もない労力がかかります。\n", "そこで、例えば京都府民から100人ランダムに選んで調査した結果から目的の割合を推測する、といった形です。\n", "\n", "これは実験に関しても同様で、「ある植物に物質Aを与えた効果」を調べる際に、複数回実験を行って効果を調べることが多いかと思います。\n", "これも、本来無限回実験を繰り返して効果を確認するところを、得られた複数回のデータから実際の効果を推測する、ということを実施している形になります。\n", "\n", "これまで、手元にあるデータの特徴を捉えるための記述統計学や可視化の方法について学んできましたが、\n", "今回は、この手元にあるデータから、元となっている集団の特徴を推測する、という数理統計学の基礎となる考え方を学んでいきます。\n", "\n" ], "metadata": { "id": "2bLFQcxy824O" } }, { "cell_type": "markdown", "source": [ "## 母集団と標本\n", "\n", "\"title\"\n", "\n", "これから特徴等を知りたい集団全体のことを**母集団**と呼びます。\n", "\n", "さきほども説明した通り、この母集団について直接知ることは難しいことが多いです。\n", "\n", "* 母集団の要素が非常に大きい。(調査対象が国民全体、実験の例のように対象が無限大の場合も。)\n", "* 予算などの制約により、調査できる数に限りがある。\n", "\n", "といったことが主な理由となります。(ちなみに集団全体を調査する場合を**全数調査**と呼ぶ。)\n", "\n", "この場合、母集団から一部を抽出し、部分的な集団を分析することで、母集団について推測を行うことになります。\n", "\n", "母集団から抽出することを**標本抽出:sampling**、抽出されたものを**標本:sample**と呼びます。\n", "\n", "統計的推測においては、抽出した標本から母集団の特徴(例えばある地域に生息する植物の草丈の特徴など)を知ることが目標となりますが、\n", "\n", "より具体的にいうと、母集団の従う**分布**を知ることが目的となります。\n", "\n", "\"title\"\n", "\n", "母集団が上図の様な分布に従っている場合、この集団から一部を標本として抽出することで、元の分布を推測する形です。\n", "\n", "そのため、統計的推測について学ぶためには分布とはいったい何なのか、\n", "\n", "つまり、**確率分布**の概念や**確率変数**について知っておく必要があります。\n", "\n", "そこで、まずは確率変数や確率分布について学習します。" ], "metadata": { "id": "7zlPZZLrvqBB" } }, { "cell_type": "markdown", "source": [ "## 確率変数と確率分布\n", "\n", "**確率**とは、事象の起こりやすさを定量的に示したものになります。事象Aの起きる確率を$P(A)$と表します。\n", "\n", "たとえば、宝くじに当たるという事象が起きる確率は2000万分の1なので、$P(宝くじに当たる) = 1/20000000$という形になります。\n", "\n", "
\n", "\n", "ここで、ある島に生息するシロツメクサの葉の長さを調査することを想定します。\n", "\n", "島に生息するシロツメクサ全個体が母集団となり、採集した個体が標本になります。\n", "\n", "ここで、採集した個体の葉の長さ$X$は、さまざまな値をとる**変数**です。\n", "\n", "この$X$の値は、ランダムな値をとるのではなく、母集団における$X$の分布をある程度反映することが想定されます。\n", "\n", "(7cmの個体が島に多く生息していれば、採集個体も7cmの長さが多くなる)\n", "\n", "\"title\"\n", "\n", "つまり、7㎝の個体が採集される事象は、母集団に7cmの個体が含まれる割合に依存して出現します。\n", "\n", "このように、とりうる各値に対し、それぞれ確率が与えられている様な変数を**確率変数**と呼びます。\n", "\n", "また、各値が観察される確率の分布を**確率分布**と呼びます。\n", "\n", "他の例でいうと、サイコロを振って出る目$X$も確率変数です。$X$は1から6までの数字を取り、各数字が生じる確率は1/6になります。\n", "\n", "$P(X=1)=1/6, P(X=2)=1/6, ... , P(X=6)=1/6$\n", "\n", "
\n", "\n", "ここで、サイコロの目$X$は1,2,3,4,5,6と、取りうる値が6種類に決まっていますが、シロイヌナズナの葉の長さは連続的な値をとります。\n", "\n", "\"title\"\n", "\n", "サイコロのような、取りうる値が可算集合(数え上げることができる集合)の値をとる確率変数$X$は**離散型の確率変数**と呼び、\n", "\n", "葉の長さの様な取りうる値が連続的な確率変数$X$は**連続型の確率変数**と呼びます。\n", "\n", "離散型の確率変数$X$の場合、それぞれの値の確率は、\n", "\n", "$P(X=x_k) = f(x_k)$ $(k=1,2,...)$ と表され、**確率分布**$f$は\n", "\n", "$f(x_k) \\geqq 0$ かつ $\\sum_{k=1}^\\infty f(x_k)=1$ の条件を満たします。\n", "\n", "サイコロの場合、$x_1 = 1, x_2 = 2, ... x_6 = 6$で、$f(x_k) = 1/6$ということになります。\n", "\n", "\"title\"\n", "\n", "連続型の確率変数$X$の場合、確率は\n", "\n", "$P(a \\leqq X \\leqq b) = \\int_{a}^{b} f(x)dx$ と表され、$f(x)$は**確率密度関数**と呼び、\n", "\n", "$f(x) \\geqq 0$ かつ $\\int_{-\\infty}^\\infty f(x)dx=1$\n", "\n", "の条件を満たします。\n", "\n" ], "metadata": { "id": "giCOeYDN5jLR" } }, { "cell_type": "markdown", "source": [ "### 確率変数の期待値と分散\n", "\n", "### 期待値\n", "\n", "確率変数は様々な値をとりますが、それらの値を代表する平均値が期待値となります。\n", "\n", "宝くじを例にすると、1/20000000の確率で1億円があたり、それ以外は100円があたる宝くじがあったとします。\n", "\n", "これは、\n", "\n", "* $P(X = 100000000) = 1/20000000$\n", "* $P(X = 100) = 19999999/20000000$\n", "\n", "と表されるので、この時この宝くじで得られるであろう金額は、\n", "\n", "$100000000 \\times 1/20000000 + 100 \\times 19999999/20000000$と、得られる金額 x その金額が起きる確率の和で計算できます。\n", "\n" ], "metadata": { "id": "S9oFCdW6j0Jc" } }, { "cell_type": "code", "source": [ "# 宝くじの期待値の計算\n", "100000000*1/20000000+100*19999999/20000000" ], "metadata": { "id": "ybDS-ZaxltTm", "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "outputId": "01b37122-2615-4881-d190-1c927b3c25d2" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "104.999995" ], "text/markdown": "104.999995", "text/latex": "104.999995", "text/plain": [ "[1] 105" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "このような、確率変数$X$に対して、それがとる値の重み付き平均を確率変数$X$の**期待値**といい$E(X)$と書きます。\n", "\n", "$E(X)$は、\n", "\n", "* 離散型の場合 $E(X) = \\sum\\limits_{x} xf(x)$\n", "* 連続型の場合 $E(X) = \\int_{-\\infty}^\\infty xf(x)dx$\n", "\n", "と定義されます。\n", "\n", "\"title\"\n", "\n", "(https://www.netkeiba.com/より引用)\n", "\n", "これはとある日の競馬の1レースのオッズ(単勝)と人気順の一部ですが、\n", "\n", "例えば人気順に勝率が\n", "\n", "* 1番人気: 40%\n", "* 2番人気: 20%\n", "* 3~6番人気: 10%\n", "\n", "だとすると、1番人気の馬に賭けて得られるお金の期待値は\n", "\n", "$0.4 \\times 2.1倍 + 0.6 \\times 0倍 = 0.84倍$\n", "\n", "と計算できます。ほかの馬についても計算してみると…\n" ], "metadata": { "id": "UoK5IeIjmBca" } }, { "cell_type": "code", "source": [ "# 期待値の計算\n", "3.4 * 0.2\n", "5.2 * 0.1\n", "13.5 * 0.1\n", "17.1 * 0.1\n", "17.7 * 0.1" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 104 }, "id": "j1M6kl4kq-7u", "outputId": "c8b049ee-e086-4daf-e748-9569a57b8e83" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "0.68" ], "text/markdown": "0.68", "text/latex": "0.68", "text/plain": [ "[1] 0.68" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/html": [ "0.52" ], "text/markdown": "0.52", "text/latex": "0.52", "text/plain": [ "[1] 0.52" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/html": [ "1.35" ], "text/markdown": "1.35", "text/latex": "1.35", "text/plain": [ "[1] 1.35" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/html": [ "1.71" ], "text/markdown": "1.71", "text/latex": "1.71", "text/plain": [ "[1] 1.71" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/html": [ "1.77" ], "text/markdown": "1.77", "text/latex": "1.77", "text/plain": [ "[1] 1.77" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "と期待値だけで見ると、5, 6番人気に賭けるのが最適となります。\n", "\n", "また、期待値の演算として重要な性質として\n", "\n", "* $E(c) = c$\n", "* $E(X + c) = E(X) + c$\n", "* $E(cX) = cE(X)$\n", "* $E(X+Y) = E(X) + E(Y)$\n", "\n", "という性質があります。" ], "metadata": { "id": "55YVtyrJyCYm" } }, { "cell_type": "markdown", "source": [ "### 分散と標準偏差\n", "\n", "記述統計学を学習した際に、データのばらつきを示す分散や標準偏差という概念を学習しました。\n", "\n", "確率変数$X$についてもばらつきの概念が存在します。\n", "\n", "例えば、\n", "\n", "さいころを1個ふった時の目を$X$、さいころを2個ふった時の目の平均値を$Y=(X_1+X_2)/2$とすると、\n", "\n", "$X$と$Y$の各値が出る確率は下図のようになります。\n", "\n", "\"title\"\n", "\n", "ここで、それぞれの期待値を求めてみると、\n", "\n", "$E(X) = 1\\times\\dfrac{1}{6}+2\\times\\dfrac{1}{6}\\cdots6\\times\\dfrac{1}{6}=\\dfrac{7}{2}$\n", "\n", "$E(Y) = E\\{(X_1 + X_2) / 2\\} = \\{E(X_1) + E(X_2)\\} / 2 = \\dfrac{7}{2}$\n", "\n", "と等しくなります。しかし、$X$の確率分布は1,2,...,6まで一様にばらついているのに対し、$Y$の確率分布は期待値に近いところに偏っています。\n", "\n", "このように、確率変数の性質は期待値だけではなく、ばらつきもとらえる必要があることがわかります。\n", "\n", "ばらつきを表すには、期待値からのずれの大きさ$X - E(X)$を考えます。\n", "\n", "ずれの大きさの平均(期待値)を求めてみると、$E(X - E(X)) = E(X) - E(X) = 0$となってしまい、正負が打ち消されてあまり意味がないので\n", "\n", "ずれの値を2乗した$(X - E(X))^2$を代わりにばらつきの指標として扱います。\n", "\n", "$(X - E(X))^2$の期待値$E\\{(X - E(X))^2\\}$を、確率変数$X$の**分散**$V(X)=E\\{(X - E(X))^2\\}$と定義しています。\n", "\n", "分散$V(X)$の計算は、\n", "\n", "* 離散型の場合$V(X) = \\sum\\limits_{x} (x - E(X))^2f(x)$\n", "* 連続型の場合$V(X) = \\int_{-\\infty}^\\infty (x-E(X))^2f(x)dx$\n", "\n", "で計算できます。\n", "\n", "※ 期待値を利用して、$V(X) = E(X^2) - 2E(X)\\times E(X) + (E(X))^2 = E(X^2) - (E(X))^2$ と計算することも可能。\n", "\n", "これら期待値や分散は、確率変数の性質を説明する際に使用されます。\n", "\n", "
\n", "\n", "実際にサイコロの出る目$X$の分散を計算してみると、期待値$E(X) = 7/2$だったので、" ], "metadata": { "id": "fhOpfqLOofiN" } }, { "cell_type": "code", "source": [ "# サイコロの出る目の分散\n", "x <- c(1, 2, 3, 4, 5, 6)\n", "sum((x - 7/2)^2 * 1/6)" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "id": "2XJysIszzdIp", "outputId": "14bbefd1-1bcf-4c56-b2a4-2a7a47cb708b" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "2.91666666666667" ], "text/markdown": "2.91666666666667", "text/latex": "2.91666666666667", "text/plain": [ "[1] 2.916667" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "と計算できます。\n", "\n", "また、分散の演算として重要な性質として\n", "\n", "* $V(c) = 0$\n", "* $V(X + c) = V(X)$\n", "* $V(cX) = c^2V(X)$\n", "* $V(X+Y) = V(X) + V(Y)$\n", "\n", "という性質があります。" ], "metadata": { "id": "-1F_xUnV2WXd" } }, { "cell_type": "markdown", "source": [ "## 代表的な確率分布\n", "\n", "サイコロの出る目$X$やさいころを2個ふった時の目の平均値を$Y$の分布をみると分かる通り、確率変数は様々な形の確率の分布に従います。\n", "\n", "どの確率変数も同じような確率で出現するものや、平均値付近の値がよく出現する場合など…。\n", "\n", "\"title\"\n", "\n", "その中でも、特定の現象によく合致する確率分布があり、正規分布、二項分布、ポアソン分布など、いくつかの確率分布が代表的なものとして挙げられます。\n", "\n", "ここでは、この中からいくつか代表的な確率分布を紹介していきます。" ], "metadata": { "id": "S2y2tXlOAcRy" } }, { "cell_type": "markdown", "source": [ "### 正規分布\n", "\n", "まずは母集団に仮定する最も一般的な分布である正規分布について見ていきます。\n", "\n", "例えば、ある島に生息する平均7㎝くらいを示すシロツメクサの草丈を計測することを考えます。\n", "\n", "\"title\"\n", "\n", "10個体計測すると左図のようになったとします。これだけだと偶然によるところが大きく、母集団における草丈の分布を想像することが難しいです。次に100個体計測してみると、右図のように中心付近の草丈の個体の出現頻度が高く、両端に行くにしたがって出現頻度がなだらかに低下していくような、釣鐘上の出現頻度を示したとします。この出現頻度からだと、母集団における確率分布も同じような形をしていそうなことが想像できます。\n", "\n", "植物の草丈や人の身長など、生物が持つ特性の多くは、このような釣鐘状の分布が見られることが多いです。\n", "\n", "このような分布(平均値付近が高く、極端な値になるにつれ出現頻度が下がる)に対して、一般的によくあてはまる確率分布として**正規分布 normal distribution**があります。(**ガウス分布**と呼ばれることもあります。)\n", "\n", "\"title\"\n", "\n", "正規分布に従う変数が観測値$x$をとる確率密度(厳密にいうと確率では無い)は、\n", "\n", "$f(x) = \\dfrac{1}{\\sigma\\sqrt{2\\pi}}e^{\\lbrace-\\frac{(x-\\mu)^2}{2\\sigma^2}\\rbrace}$\n", "\n", "と表されます。ここで、$\\mu$は平均、$\\sigma^2$は分散を指す。この関数$f(x)$は正規分布の**確率密度関数**と呼ばれます。\n", "\n", "本講義ではこの数式を覚える必要はありませんが、\n", "\n", "正規分布は**平均と分散**が決まればどのような分布になるのかが決まる、という点は理解できるかと思います。\n", "\n", "平均$\\mu$、分散$\\sigma^2$の正規分布を$N(\\mu, \\sigma^2)$と表記し、確率変数$X$がこの正規分布に従うことを\n", "\n", "$X \\sim N(\\mu, \\sigma^2)$\n", "\n", "と表します。\n", "\n", "Rを使用することで、確率密度関数の可視化が可能です。\n", "\n", "`dnorm`関数と`ggplot2`を使用することで正規分布の確率密度関数を表すことができます。\n", "\n", "`dnorm`関数はある平均値と分散に従う正規分布の確率密度を計算することができます。\n", "\n", "```\n", "dnorm(xの値, mean=平均値, sd=標準偏差)\n", "```\n", "\n", "たとえば、ある学校の学生の身長の分布が平均$\\mu=170$, 分散$\\sigma=5^2$の正規分布$X \\sim N(170, 5)$に従っていた時、下記のようなコードで可視化できます。\n" ], "metadata": { "id": "kGgTx5SRbfFN" } }, { "cell_type": "code", "source": [ "# 正規分布の描写\n", "library(ggplot2)\n", "x <- seq(150,190,0.1) # x = 150 ~ 190の範囲を描写\n", "m <- dnorm(x,mean=170,sd=5) # 平均170, 標準偏差5の正規分布\n", "data <- data.frame(x=x,y=m)\n", "g <- ggplot(data,aes(x=x, y=y))\n", "g <- g + geom_line()\n", "g" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 437 }, "id": "1G_dzNTRbhPO", "outputId": "a78a0481-00be-4554-80e0-d9a6c6ebb159" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/plain": [ "plot without title" ], "image/png": 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/bXNdgDqTP9DEiAZD2bIEnx7K9rsAfSaJoOSIBkPZsgLaP+/FPtgPQFPQRIgGQ9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\nG6WF7QCkiEDy/w9smMp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}, "metadata": { "image/png": { "width": 420, "height": 420 } } } ] }, { "cell_type": "markdown", "source": [ "平均値や分散を少し変えた分布を重ねてみると、下のようになります。" ], "metadata": { "id": "lN3XpBZRrPoY" } }, { "cell_type": "code", "source": [ "# 複数の正規分布を描写\n", "library(ggplot2)\n", "x <- seq(150,190,0.1) # x = 150 ~ 190の範囲を描写\n", "m0 <- dnorm(x,mean=170,sd=5) # 平均170, 標準偏差5の正規分布1\n", "m1 <- dnorm(x,mean=180,sd=5) # 平均180, 標準偏差5の正規分布2\n", "m2 <- dnorm(x,mean=170,sd=10) # 平均170, 標準偏差10の正規分布3\n", "data <- data.frame(x=x, y0=m0, y1=m1, y2=m2)\n", "g <- ggplot(data,aes(x=x))\n", "g <- g + geom_line(aes(y=y0)) # 正規分布1を描写\n", "g <- g + geom_line(aes(y=y1), color=\"red\") # 正規分布2を赤線で描写\n", "g <- g + geom_line(aes(y=y2), color=\"blue\") # 正規分布3を青線で描写\n", "g" ], "metadata": { "id": "IaG2_lITrDZl", "colab": { "base_uri": "https://localhost:8080/", "height": 437 }, "outputId": "18101723-71e7-4c65-a16c-ee205e4fa242" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/plain": [ "plot without title" ], "image/png": 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qHNd0CWZY7jLvZfzXw8tyiKbW1tyb7K0KFDJUk6c+ZMsi+kW06nU5bliIX/HzEj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}, "metadata": { "image/png": { "width": 420, "height": 420 } } } ] }, { "cell_type": "markdown", "source": [ "\"title\"\n", "\n", "ここで、母集団において、$x$以下の値を持つ個体の割合は、\n", "\n", "$F(x) = P(X \\leq x) = \\int_{-\\infty}^{x}f(y)dy$として得られ、この関数$F(x)$は**分布関数**と呼びます。\n", "\n", "Rでは、`pnorm`関数で計算することができます。\n", "\n", "```\n", "pnorm(xの値, mean=平均値, sd=標準偏差)\n", "```\n", "\n", "例えば先ほどの身長の平均$\\mu=170$, 分散$\\sigma=5^2$の正規分布$X \\sim N(170, 5)$で、183以下の確率は下記のように計算できます。" ], "metadata": { "id": "w56QmAgJ1Ikl" } }, { "cell_type": "code", "source": [ "# 平均170,標準偏差5の正規分布における183以下の確率\n", "pnorm(183,170,5)" ], "metadata": { "id": "OYQSVPxO4ohZ", "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "outputId": "0b765fb0-225e-46f4-bfc8-fdd49329f250" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "0.995338811976281" ], "text/markdown": "0.995338811976281", "text/latex": "0.995338811976281", "text/plain": [ "[1] 0.9953388" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "図として表すと、下図の様になります。" ], "metadata": { "id": "g79mq7dCg3rB" } }, { "cell_type": "code", "source": [ "# 183以下の領域を可視化\n", "library(ggplot2)\n", "\n", "x <- seq(150,190,0.1)\n", "y <- dnorm(x, 170, 5)\n", "data <- data.frame(x=x, y=y)\n", "\n", "x_color <- seq(150,183,0.1)\n", "y_color <- dnorm(x_color, 170, 5)\n", "data_color <- data.frame(x=x_color, y=y_color)\n", "\n", "g <- ggplot()\n", "g <- g + geom_path(data=data, aes(x=x, y=y))\n", "g <- g + geom_ribbon(data=data_color, aes(x=x, y=y, ymin=0, ymax=y))\n", "g" ], "metadata": { "id": "eXIHmoVy5PkL", "colab": { "base_uri": "https://localhost:8080/", "height": 437 }, "outputId": "8715a91b-7605-485c-c468-9330fa5ba58d" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/plain": [ "plot without title" ], "image/png": 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cUACUgC2YC0sGPEkCYWWw2QgCSQDUiR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\nFCFSo3JGCaQ0Q7rK9l5Mdt/PGSWQ0gxpP9tbMdmpl7eOEkgphvQ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9Z87K3Y+N50up0xOQkt6Krt2cwTsssL2MtAekxDdVjWh7p+1FpD4g\nJb9B6jjbSyAgJb66o9rsucr2IlIfkBLfje2f3HGQ7UWkPiAlvY86DXNuU1NsLyPtASnhbTp8\n77VO3RHfXG57ISkPSAnvBpV9n/L3OpTbXkjKAxKRQEAiEghIRAIBiUggIBEJBCQigYBEJBCQ\niAQCEpFAQCISCEhEAgGJSKD/B1s5cmWS5eKbAAAAAElFTkSuQmCC" }, "metadata": { "image/png": { "width": 420, "height": 420 } } } ] }, { "cell_type": "markdown", "source": [ "では、下図の様な165以上180以下の身長の確率はどう計算すればよいでしょうか？\n", "\n", "\"title\"\n", "\n", "これも`pnorm`関数を用いることで計算できます。" ], "metadata": { "id": "hJL7iQdy6DQ-" } }, { "cell_type": "code", "source": [ "# 165~180の範囲の確率を求める\n", "pnorm(180,170,5) - pnorm(165,170,5)" ], "metadata": { "id": "LFqHtkww6IuO", "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "outputId": "85528e3d-f13d-4dcc-8f6f-b94b67a0bb2a" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "0.818594614120364" ], "text/markdown": "0.818594614120364", "text/latex": "0.818594614120364", "text/plain": [ "[1] 0.8185946" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "この様に、正規分布に従う変数$X$について、`pnorm`関数を使用することで、得られた観測値$x$がある値の範囲内である確率を計算することができます。\n", "\n", "また、ある正規分布に従う母集団から無作為に抽出した結果をシミュレートすることもできます。\n", "\n", "Rの`rnorm`関数で実施できます。\n", "\n", "```\n", "rnorm(n=抽出数, mean=平均値, sd=標準偏差)\n", "```\n", "\n", "平均$170$、分散$5^2$の正規分布から5標本無作為に抽出してみると…\n", "\n", "(無作為抽出なので動かすごとに結果は変わります。)" ], "metadata": { "id": "Bc0YtGXs9TWV" } }, { "cell_type": "code", "source": [ "# 平均170 標準偏差5の正規分布から5標本をランダムに抽出する\n", "rnorm(n=5, mean=170, sd=5)" ], "metadata": { "id": "AEaO3YTtBkgA", "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "outputId": "37790636-52cb-46b2-b4a0-67afb7bb88b5" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "\n", "
  1. 170.65229671653
  2. 168.375173912678
  3. 171.778591588002
  4. 164.512905411614
  5. 167.715986459784
\n" ], "text/markdown": "1. 170.65229671653\n2. 168.375173912678\n3. 171.778591588002\n4. 164.512905411614\n5. 167.715986459784\n\n\n", "text/latex": "\\begin{enumerate*}\n\\item 170.65229671653\n\\item 168.375173912678\n\\item 171.778591588002\n\\item 164.512905411614\n\\item 167.715986459784\n\\end{enumerate*}\n", "text/plain": [ "[1] 170.6523 168.3752 171.7786 164.5129 167.7160" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "といった形で抽出できます。\n", "\n", "5標本抽出した結果からヒストグラムを描いてみると" ], "metadata": { "id": "5qO_y9BbBoG0" } }, { "cell_type": "code", "source": [ "# 平均170 標準偏差5の正規分布から5標本をランダムに抽出し、ヒストグラムを描く\n", "hist(rnorm(n=5, mean=170, sd=5))" ], "metadata": { "id": "2KOaQ24pB4fn", "colab": { "base_uri": "https://localhost:8080/", "height": 437 }, "outputId": "66548217-d6d3-4a45-fffa-d52bb1380610" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/plain": [ "Plot with title “Histogram of rnorm(n = 5, mean = 170, sd = 5)”" ], 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EmKkHwJfX6EJEVIvoQ+P0KSIiRfQp8fIUkRki+hz4+QpAjJl9DnR0hShORL6PMj\nJClC8iX0+RGSFCH5Evr8CEmKkHwJfX6EJEVIvoQ+P0KSIiRfQp8fIUkRki+hz4+QpAjJl9Dn\nR0hShORL6PMjJClC8iX0+RGSFCH5Evr8CEmKkHwJfX6EJEVIvoQ+P0KSIiRfQp8fIUkRki+h\nz4+QpAjJl9DnR0hShORL6PMjJClC8iX0+RGSFCH5Evr8CEmKkHwJfX6EJEVIvoQ+P0KSIiRf\nQp8fIUkRki+hz4+QpAjJl9DnR0hShORL6PMjJClC8iX0+RGSFCH5Evr8CEmKkHwJfX6EJEVI\nvoQ+P0KSIiRfQp8fIUkRki+hz4+QpAjJl9DnR0hShORL6PMjJClC8iX0+RGSFCH5Evr8CEmK\nkHwJfX6EJEVIvoQ+P0KSIiRfQp8fIUkRki+hz4+QpAjJl9DnR0hShORL6PMjJClC8iX0+RGS\nFCH5Evr8CEmKkHwJfX6EJEVIvoQ+P0KSIiRfQp8fIUkRki+hz4+QpAjJl9DnR0hShORL6PMj\nJClC8iX0+RGSFCH5Evr8CEmKkHwJfX6EJEVIvoQ+P0KSIiRfQp8fIUkRki+hz4+QpAjJl9Dn\nR0hShORL6PMjJClC8iX0+RGSFCH5Evr8CEmKkHwJfX6EJEVIvoQ+P0KSIiRfQp8fIUkRki+h\nz4+QpAjJl9DnR0hShORL6PMjJClC8iX0+RGSFCH5Evr8CEmKkHwJfX6EJEVIvoQ+P0KSIiRf\nQp8fIUkRki+hz4+QpAjJl9DnR0hShORL6PMjJClC8iX0+RGSFCH5Evr8CEmKkHwJfX6EJEVI\nvoQ+P0KSIiRfQp8fIUkRki+hz4+QpAjJl9DnR0hShORL6PMjJClC8iX0+RGSFCH5Evr8CEmK\nkHwJfX6EJEVIvoQ+P0KSIiRfQp8fIUkRki+hz4+QpAjJl9DnR0hShORL6PMjJClC8iX0+RGS\nFCH5Evr8CEmKkHwJfX6EJEVIvoQ+P0KSIiRfQp8fIUkRki+hz4+QpAjJl9DnR0hShORL6PMj\nJClC8iX0+RGSFCH5Evr8CEmKkHwJfX6EJEVIvoQ+P0KSIiRfQp8fIUkRki+hz4+QpAjJl9Dn\nR0hShORL6PMjJClC8iX0+RGSFCH5Evr8CEmKkHwJfX6EJEVIvoQ+P0KSIiRfQp8fIUkRki+h\nz4+QpAjJl9DnR0hShORL6PMjJClC8iX0+RGSFCH5Evr8CEmKkHwJfX6EJEVIvoQ+P0KSIiRf\nQp8fIUkRki+hz4+QpAjJl9DnR0hShORL6PMjJClC8iX0+RGSFCH5Evr8CEmKkHwJfX6EJEVI\nvoQ+P0KSIiRfQp8fIUkRki+hz4+QpAjJl9DnR0hShORL6PMjJClC8iX0+RGSFCH5Evr8CEmK\nkHwJfX6EJEVIvoQ+P0KSIiRfQp8fIUkRki+hz4+QpAjJl9DnR0hShORL6PMjJClC8iX0+RGS\nFCH5Evr8CEmKkHwJfX6EJEVIvoQ+P0KSIiRfQp8fIUkRki+hz4+QpAjJl9DnR0hShORL6PMj\nJClC8iX0+RGSFCH5Evr8CEmKkHwJfX6EJEVIvoQ+P0KSIiRfQp8fIVWp9ZXZ99zzyOsdnIuQ\nfAl9foRUlcWnDXYpI7+/vNL5CMmX0OdHSNVYuIX7xNSZl1xyzuShbvvFFc5ISL6EPj9Cqsb0\nxjszWy1X1p1S4YyE5Evo8yOkagyZ1rZ9yIgKZyQkX0KfHyFVo/HCtu3zuhd98tVBA3P6utXt\n7GJ6Y7+g1TcNDFpT4PPrFvj6Gqev7Y2/jLUOadTBbduTNi/65Jo5s3MeuqW9XSycHbY77qj1\nCipjfV20cG1v/GWsdUin1F26Mr217Htuhmo5wPpprUNasoPru9fUE0+Yskcvt9tS5ZKA9c/a\n/xxp1Y9H1yc/Rmrc+doW4YKA9VGXfkVoxV/nzWtu7zU5wBD/v2sHGEBIgAAhAQKEBAgQEiBA\nSIAAIQEChAQIEBIgQEiAACEBAoQECBASIEBIgAAhAQKEBAgQEiBQy5B2dkAN7Sy8MdcypMMm\nzg3aRNbXJcGv7zDhjbmWIU0N/f9kyvq6xNT6CKl9rK9rTK2PkNrH+rrG1PoIqX2sr2tMrY+Q\n2sf6usbU+gipfayva0ytj5Dax/q6xtT6CKl9rK9rTK2PkNrH+rrG1PoIqX2sr2tMra+WIR19\ndA0PXgXW1zWm1lfLkBYvruHBq8D6usbU+vhnFIAAIQEChAQIEBIgQEiAACEBAoQECBASIEBI\ngAAhAQKEBAgQEiBASIAAIUwOTPwAAApeSURBVAEChAQIEBIgsK5DWn1Gtx1TGzdk/iLADwo2\nay63vih6YPc+/b88J9lacsqoxs2mL6zdqtqUW1+Q82vK/smH18KcX976JPNbxyEt2KFv5opc\n5ibPSDxasFlrbeuLrndbnXP6oO5PRNGqHdxBF05r3CKAf/BZdn1Bzu+c1JJmbN7jgzDnl7c+\nyfzWbUgf9hzb3JS+IjPdc9mP5m3WWN763ukzZlkUNfc5Pop+7H4Yf+AOd1ptFxe1t74g55c2\nt/6CQOeXllqfZH7rNqQPTlsdZa7IKa45+9G8zRrLW9+l7sHkpDX+b3Tflcnm1oNba7eytPLr\nC3J+KS1jPr0q0PmlpNcnmd+6f7Ehc0WmuPda3niveDMAmfWN77k6WvlhsrWifq/UR6a6V2q3\nqjYl6wtzfimXuTmhzi8ltT7N/GoW0gHu7IHOffLWws0AZNY3atvnd6lzW90QRX916f//2Uw3\nu4bLyilZX5jzSywblCQU5vwS6fVp5lezkPZwW8666cx+7pqCzQBk1td31Gan3X35SHdrNM+d\nkPrIpe6eWq4rq2R9Yc4vcbH7Q/w2zPkl0uvTzK9mIT1yd/xcOXqpaaNV+ZsBaMq+PPrL+O3C\nPkNa5rkTUx+5xP2mhsvKKVlfmPOLLd9k9+QkzPlFufVp5lezkDIOdM+W2aylzPo2rv8oOfmG\n+99mNyX1kXPcwzVbVJ6S9WU/Edb8YrekWo/CnF+UW19W1+ZX65COcY+W2aylzPp2rF+dnBzv\nnljVsEfqI5Pd32u3qjYl68t+Iqz5xSbWL0lOwpxflFtfVtfmV6uQll51W+q9Xd0reZvrfDFl\nZAZ9ons6OdnbvR7t1Cv55r9m6IhaLiunZH1hzi8OqPfY9EaQ82tbn2Z+tQppzbA+f45PfuvG\n5G+GIDPouXV7royi57p9LoqudefFH7janV/bhWWUrC/M+UXRC256eiPI+bWtTzO/dRvSYzNm\nzKgfEr95P7q3rvf0cw+s6zcvyt+ssbz1Rae60ed/u2f3OVHUspubdP6hddt9VOvltbO+MOcX\n3e4uSH80zPm1rU8yv3Ub0qzsbwo2R9GT+wxoGPqt1M+U8zZrK399rdds36P/vqnnn0tPH9U4\n7IQPar26dtcX5Pziu6DLMx8Ocn5561PMj39GAQgQEiBASIAAIQEChAQIEBIgQEiAACEBAoQE\nCBASIEBIgAAhAQKEBAgQEiBASIAAIQEChAQIEBIgQEiAACEBAoQECBASIEBIgAAhAQKEBAgQ\nEiBASIAAIQEChAQIEBIgQEiAACEBAoQECBCSXyd0n1vrJQjU71TNuc5tfMz3QsJFSF7d5n5S\n1fluyPxJxh/4W8rqM7ql/wxxU/bvP74WRUtOGdW42fSFHV22UkhtS2/ZffC7qtWudwjJp6Ub\n71zdGS9zk2ckHvW2lAU79M2EdE7qSDM27/FBtGoHd9CF0xq3WNzBhSuFlLf05vppugWvZwjJ\np4vd/dWdcaZ7zu9KPuw5trlpx7wPzK2/IIp+7H4Yb97hTuvg0pVCyl/6YQ2vdmWR6zNCqs4h\n7p1xPe6NJrul3x3VffiPW+MP/W3q0MaNJz6T/8klRw/uudMzH50ytPcXk781v2bIp5LL5l8o\npTnr7exHTnFl/6Z2yR6jRcePbNxkUupvmT9zwMaNo454rboDfHDa6ig/pJYxn14VRaP7rkze\n2Xpwa5Rv5SWf69dnu0vWxJv379Bj0PQlbSFVXvpcd3I1w9wQEVJ1vukO2+eiF6MpbvyxTz2x\nt7s+il4f3Oc7N144rOnxvE+OO//5G3uM3G/G3LsHbLo6ip5L367yLpTycfZJipuU3f0U917L\nG++VHLVkj++O6j/j5ouGN8XP6uf2GPr9a8/oO/j9qg6QyA/pMjcnilbU75V6Z6p7peCwR7rD\nrr7mQHdCFP2xfuhFPz9it8ZsSB0svXXQ1p2e7AaCkKozze2dfIOe7ibHb19x+yU3oHvizQX1\nO+d98rj47cHu61HyffqJKJrlfhsVXiil9a6sJ7O7P8CdPdC5T95adNSSPR7XkDyOer3v2Ci6\naoc58eYV7oqqDpDIC2nZoCShv7qpqfdmutkFh+31xeTtvx7UEu3jkju/4102pI6WfkjyCoZJ\nhFSd6e7W9MmDyUmv0VFr/01Tj4d2de+3fTK5PZ7tbo7fXuXuTgKbX3ihdu3htpx105n93DXF\nRy3cY+smO7ydGO+Wpj6/esUjyfObKg6QyAvpYveH+O285E4ndmnqe0Kb/kPfSW+s6blVcvKC\nq/AcqWDp5xQlaQchVWe6m5s+WZCc9P9MtNDtmfnEk4WfnOmSl95+7n4VRfu7RYUXatcjdy+L\n377UtNGqoqMW7nFR7pHVS1F00+4Dkq1TqjpAoi2k5ZvsnpzMcyem3r3E/abgjJe7ft+8/s14\n4033leT9FZVCKlj6T5KrbRIhVWd6+il15iS+yTa7ialPnBh/Dy745EwXP2tKh7SHW1F4oYzS\nZ+wZB6YeSRUftW2PzW7079KWRGe6sTc89tQv0iFVdYC2kG5xv0ydzU1JvXuOe7hwHY8c0NvV\n7fu3+LFf+krWVXixoWDpNxXfp5pBSNUpCentzD3Ske7pdkPK3SMV3s7beS0gdowr/EFS8R4X\nudyjtxU9RyQP7x4sF1KHLzZMrF+SnKxq2CP17mT39+Lru3L2lLqtV72Rvkda6iq82FCwdO6R\nUFlJSNFGm6WeI+1Ut6TdkKYlj8BKb+elz9iXXnVb6nTXopfPSva4SY9UAe9G0WvuwGTrzHIh\ndfRiw6reY9MbO/X6KH67ZuiIclf5OPfMx91Tr8I9UeHFhsKln8tzJFRUGtJRqScWL9TtFbUb\n0ix3b9GF2rFmWJ8/xye/dWPiW/kLbT+XKdnjce6sePPdIftFy+vGJIcf5o6p5gApuZBecNPT\nG9e68+K3V7vzCw771NDUA78T3PPxg9PkAdthFZ4j5S89ig7lVTtUVBrSW0P6nPXL8wf3/VP7\nIT2b3F1UdTu/t6739HMPrOs3L3niskvRUfP2+M5Id+SNF41sfCiK9nPH/OrcgQ80DL9tWRUH\neGzGjBn1Q+I378fv3O4uSH+0ZTc36fxD67b7qOCwH3+2+7evvGpat11bowfqBp9x6X579q/w\nYkPe0qPWwfwcCRWVhhS9fuRmDYMPXVD8ybyb/ZpNP118ofY8uc+AhqHfSs7X7HYrOmreHqO3\njxvRMGD/5Ncp3j1sUP89H4/O7zPk7SoOMCv73CY559Xu8syHl54+qnHYCR8UHfaDU7fq1X/7\ni5LnYLdv133QtCUjxlS19GieO6nCGTdohOTTLPdA5y90XfHz+HVDcdjDG17p+EwbJkLyaenG\nX+z8hQ66RL+QdXPYl/ntb/hR7b9HyrP8/CU+VrIODsu/R4I3J24Q/0K2Ouc2zqn1EmqHkAAB\nQgIECAkQICRAgJAAAUICBAgJECAkQICQAAFCAgQICRAgJECAkAABQgIECAkQICRAgJAAAUIC\nBAgJECAkQICQAAFCAgQICRAgJECAkAABQgIECAkQ+H+EsMSBY0NGywAAAABJRU5ErkJggg==" }, "metadata": { "image/png": { "width": 420, "height": 420 } } } ] }, { "cell_type": "markdown", "source": [ "と全く正規分布しているようには見えません。サンプルサイズが限られている場合、母集団が正規分布に従っていたとしても、標本データからそのことは分からない場合が多々あります。\n", "\n", "標本データから得られる標本分布は、サンプルサイズが小さいと、あまり意味がないことに注意が必要です。\n", "\n", "サンプルサイズが非常に大きければ標本から得られるヒストグラムは母集団の分布に近づいていきます。\n", "\n", "例えば10,000個体標本抽出をした場合を見てみましょう。" ], "metadata": { "id": "57xRgCGoB7We" } }, { "cell_type": "code", "source": [ "# 平均170 標準偏差5の正規分布から10000標本をランダムに抽出する\n", "hist(rnorm(n=10000, mean=170, sd=5), breaks=100)" ], "metadata": { "id": "gFtcF34hCz7a", "colab": { "base_uri": "https://localhost:8080/", "height": 437 }, "outputId": "0d020706-3e0f-4a91-e76e-88be36164c47" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/plain": [ "Plot with title “Histogram of rnorm(n = 10000, mean = 170, sd = 5)”" ], "image/png": 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C/7fyGbs/HirgKP6Ej+dPgXsnJdGr2L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\nrkNa31fsXj1v4cK5E6tE/1qHHZsP6Tn7rdNP/01yiXPtLwzud42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SIMH/B1e41p2seuM+AAAAAElFTkSuQmCC" }, "metadata": { "image/png": { "width": 420, "height": 420 } } } ] }, { "cell_type": "markdown", "source": [ "この様に、かなり正規分布に近いヒストグラムが得られました。\n", "\n", "ここまで、いくつか正規分布に関係するRの関数を扱ってきたのでまとめておきます。\n", "\n", "\"title\"\n", "\n", "* `dnorm` ... ある正規分布における各値の確率密度を計算出来る\n", "\n", "```\n", "dnorm(xの値, mean=平均値, sd=標準偏差)\n", "```\n", "\n", "* `pnorm` ... $x$以下の値を持つ確率、分布関数の計算が出来る\n", "\n", "```\n", "pnorm(xの値, mean=平均値, sd=標準偏差)\n", "```\n", "\n", "* `rnorm` ... 正規分布からのランダムな標本抽出が出来る\n", "\n", "```\n", "rnorm(n=抽出数, mean=平均値, sd=標準偏差)\n", "```\n", "\n", "* `qnorm` ... (初登場)パーセンタイル値、分布関数である確率になる$x$の値が求まる(`pnorm`の逆)\n", "\n", "```\n", "qnorm(確率の値, mean=平均値, sd=標準偏差)\n", "```" ], "metadata": { "id": "zmrb6jOoC9bZ" } }, { "cell_type": "markdown", "source": [ "例えば第3回で扱った偏差値を例にすると、\n", "\n", "仮にある試験の偏差値が平均値が50、標準偏差が10の正規分布$N(50, 10)$に従っていたとします。" ], "metadata": { "id": "JhzNKtxVvxs3" } }, { "cell_type": "code", "source": [ "# 偏差値の分布の描写\n", "library(ggplot2)\n", "x <- seq(0,100,0.1) # x = 0 ~ 100の範囲を描写\n", "m <- dnorm(x,mean=50,sd=10) # 平均50, 標準偏差10の正規分布\n", "data <- data.frame(x=x,y=m)\n", "g <- ggplot(data,aes(x=x, y=y))\n", "g <- g + geom_line()\n", "g" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 437 }, "id": "UKlXL6P0xXQE", "outputId": "6efcc737-b3ff-4783-80ea-dcf07d9fad8c" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/plain": [ "plot without title" ], "image/png": 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cpK/T/9qRyQyp5/HJAAiSGXIH1HbfSnckCaQkMj1gASIDHkEqS36FH9qRyQFtPd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\nTVruQi4AAAAASUVORK5CYII=" }, "metadata": { "image/png": { "width": 420, "height": 420 } } } ] }, { "cell_type": "markdown", "source": [ "この分布において、偏差値が60以下の人達の割合を求めると…" ], "metadata": { "id": "bxrxkYnYxWvU" } }, { "cell_type": "code", "source": [ "# 平均50, 標準偏差10の正規分布で、60以下の割合を求める\n", "pnorm(60, 50, 10)" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "id": "p4Q4wn9lwpag", "outputId": "b8edf1af-808a-44e0-f85b-7dc94f60c199" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "0.841344746068543" ], "text/markdown": "0.841344746068543", "text/latex": "0.841344746068543", "text/plain": [ "[1] 0.8413447" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "また、95％の人が含まれる偏差値xを求めると…" ], "metadata": { "id": "M7-VIqRJxAt6" } }, { "cell_type": "code", "source": [ "# 平均50, 標準偏差10の正規分布で、分布関数で確率が95%になるxの値を求める\n", "qnorm(0.95, 50, 10)" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "id": "g-kyBOTQxRpT", "outputId": "18817b80-1aae-4ad3-abe6-46c71d3423d7" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "66.4485362695147" ], "text/markdown": "66.4485362695147", "text/latex": "66.4485362695147", "text/plain": [ "[1] 66.44854" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "といった形で各種関数を活用します。\n", "\n", "また、これらの関数名の`norm`を別の確率分布名に入れ替えた`dxxxx`, `pxxxx`, `rxxxx`, `qxxxx`という関数が後ほど別の確率分布を扱う際に登場します。" ], "metadata": { "id": "Agztmarmxlqy" } }, { "cell_type": "markdown", "source": [ "#### 標準正規分布\n", "\n", "正規分布の中でも、平均$\\mu=0$, 分散$\\sigma=1^2$の正規分布$N \\sim N(0, 1)$を特に**標準正規分布**と呼び、$\\phi(z)$で表します。\n", "\n", "この分布は、\n", "$\\phi(z) = \\dfrac{1}{\\sqrt{2\\pi}}exp\\lbrace-\\frac{z^2}{2}\\rbrace$と表され、代表的な分布として主な区間の確率がよく知られています。\n", "\n", "例えば、$P(-1 \\leqq Z \\leqq 1)$を`pnorm`関数で求めると、\n", "\n", "\n", "\n", "\n" ], "metadata": { "id": "Gkw9_uOotdoa" } }, { "cell_type": "code", "source": [ "# 標準正規分布(平均0, 分散1)における-1から1の確率を求める\n", "pnorm(1, 0, 1) - pnorm(-1, 0, 1)" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "id": "PkAFjtSG4-K2", "outputId": "f622b365-0f28-4041-9645-2952975e7f0d" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "0.682689492137086" ], "text/markdown": "0.682689492137086", "text/latex": "0.682689492137086", "text/plain": [ "[1] 0.6826895" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "この様に、標準偏差の$k$倍の範囲をそれぞれ求めていくと下記の様な確率が求まります。\n", "\n", "* $P(-1 \\leqq Z \\leqq 1)$ ... 0.6827 (約1/3)\n", "* $P(-2 \\leqq Z \\leqq 2)$ ... 0.9545 (約1/20)\n", "* $P(-3 \\leqq Z \\leqq 3)$ ... 0.9973 (約3/1000)\n", "\n", "\"title\"\n", "\n", "これらの値から、平均値から標準偏差の3倍である$3\\sigma$を超える様な値は極めて起こりにくいことが分かります。\n", "\n", "平均$\\mu$、分散$\\sigma^2$の正規分布$N(\\mu, \\sigma^2)$に従う変数$X$は、$Z = \\dfrac{X-\\mu}{\\sigma}$という変数の変換を行うことで、\n", "\n", "標準正規分布に従う変数$Z \\sim N(0, 1)$となります。この変換を**標準化**と呼びます。\n", "\n", "どんな正規分布でも標準化することで平均0、標準偏差1の標準正規分布に変換できるので、異なる平均と標準偏差を持つ正規分布のデータを比較したり、上の範囲などをもとに確率を捉えたりすることが容易になります。\n" ], "metadata": { "id": "e0GW2OQj4-WI" } }, { "cell_type": "markdown", "source": [ "### 二項分布\n", "\n", "続いて、離散型の確率変数の代表的な分布である**二項分布**について触れます。\n", "\n", "\"title\"\n", "\n", "例えば、ある地域にはシロツメクサの中でも白色のものと薄紅色のものが生息しています。薄紅色のものが20%($p=0.2$)、白色のものが80%($q=0.8$)生息しているとします。\n", "\n", "この時、無作為に4個体シロツメクサを採集すると、薄紅色のシロツメクサの個体数Xは0,1,2,3,4のいずれかの値をとる確率変数になります。\n", "\n", "\"title\"\n", "\n", "それぞれが起きる確率としては、\n", "* $P(x=0) = _4C_0p^0q^4=1 \\times 0.8^4 = 0.4096$\n", "* $P(x=1) = _4C_1p^1q^3=4 \\times 0.2 \\times 0.8^3 = 0.4096$\n", "* $P(x=2) = _4C_2p^2q^2=6 \\times 0.2^2 \\times 0.8^2 = 0.1536$\n", "* $P(x=3) = _4C_3p^3q^1=4 \\times 0.2^3 \\times 0.8 = 0.0256$\n", "* $P(x=4) = _4C_4p^4q^0=1 \\times 0.2^4 = 0.0016$\n", "\n", "となり、$X$の取りうる値全ての確率の和$\\sum_{k=0}^4 P(X=k)$が$1$になっています。\n", "\n", "\"title\"\n", "\n", "この様な、母集団がＡとＢの２種類に分けられ、それぞれの種類に属する要素の割合が$p$および$q=1-p$であるとき、\n", "\n", "大きさ$n$から成る標本集団にＡの標本が$x$個含まれる確率は\n", "\n", "$f(x) = _nC_xp^x(1-p)^{n-x}=\\dfrac{n!}{x!(n-x)!}p^x(1-p)^{n-x}$\n", "\n", "と表せます。\n", "\n", "正規分布の際はこのような式を確率密度関数と呼びましたが、離散型確率変数の場合は**確率関数**と呼びます。\n", "\n", "この確率関数に従う分布を**二項分布binomial distribution**と呼び$Bi(n, p)$と表します。\n", "\n", "二項分布$Bi(n, p)$に従う確率変数$X$の平均(期待値)と分散は、\n", "\n", "$E(X) = np$\n", "\n", "$V(X) = np(1-p)$\n", "\n", "で表されます。\n", "\n", "平均に関しては、$p$の確率で出現する個体は$n$個体サンプリングすれば$n\\times p$個体くらいは出てきそう、と直感に合う形です。\n", "\n", "分散についても、$p=\\dfrac{1}{2}$(50%)の時に一番バラバラな結果が得られそう(分散が大きい)だと直感的に理解できます。\n" ], "metadata": { "id": "vzGFVshQmTe8" } }, { "cell_type": "markdown", "source": [ "正規分布同様、二項分布もRの関数で確率などの計算が容易にできます。\n", "\n", "`dbinom`関数で確率が$p$の事象を$n$回試行する二項分布$Bi(n, p)$において、その事象が$x$回起きる確率を計算出来ます。\n", "\n", "```\n", "dbinom(x, size=試行回数, prob=事象の確率)\n", "```\n", "\n", "例えば、80%の確率で勝つ勝負を10回繰り返したときに、10回勝つ事が出来る確率は..." ], "metadata": { "id": "-MpubuxYSgXv" } }, { "cell_type": "code", "source": [ "# n=10, p=0.8の二項分布でx=10の確率を求める\n", "dbinom(10, size=10, prob=0.8)" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "id": "G137cfgvTfoh", "outputId": "440ca769-4da5-45b8-a830-6401d42b9a9e" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "0.1073741824" ], "text/markdown": "0.1073741824", "text/latex": "0.1073741824", "text/plain": [ "[1] 0.1073742" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "`dbinom`関数で1 ~ 10回それぞれの確率を計算してグラフ化してみると下図のようになります。" ], "metadata": { "id": "9PQ2h4boVFxQ" } }, { "cell_type": "code", "source": [ "# 二項分布の描写\n", "library(ggplot2)\n", "x <- seq(1,10,1) # x = 1 ~ 10の範囲を描写\n", "m <- dbinom(x,size=10,prob=0.8) # 試行回数10, 確率0.8の二項分布\n", "data <- data.frame(x=x,y=m)\n", "g <- ggplot(data,aes(x=x, y=y))\n", "g <- g + geom_bar(stat = \"identity\", position = \"dodge\", fill = \"#00A968\") # 棒グラフを追加\n", "g <- g + theme(text = element_text(size = 18)) # フォントサイズ変更\n", "g <- g + scale_x_continuous(breaks = seq(1,10,1)) # 軸ラベル追加\n", "g" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 437 }, "id": "-cpWApAPVLR-", "outputId": "d8d75e49-24f2-45ab-d333-404329f84abd" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/plain": [ "plot without title" ], "image/png": 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8laIJ0RkIBktFHybzOskLfnv+87fEQy\nAhKQjJbJW703Lpfn572r/Rl5FEhn9AGDlHgFA+ncaooc6r11tTw17333y8bw/e+p48ePTx0u\na6r83Z3ajdgRsyciBxzJ6rGN+tHIAR1IsY3mu5EDTmaTsY1W7IAOpMgRR+ZiG7NZ7Nken+38\nJekKPnx4cjpxorMxPZn8MA7Pxl6wo9ls7BVrHCkfSIE0LXKs99YWeVy/64GRC98Jb6xZtGjR\n6grbdD6UdAUPYGIQG+/JC3XqWwSVILXO9hGpebtc9GbvzS9t2LDhi3NlNZul7+6UtWNHtBqR\nAxpZK7oRO+AcNqIPtJlFn+05bMSebXyjFd1odJ9s0hXcO7VJE4PYmBvM5VH+ktaTIGXLZX/v\njcvkhcLdk9fK5QeLx5V+8snXSDq+RlIbw/A1UnZl7qc9Ju8UHF0h102r40ofIZB0QFIbQwHp\ndtkVbvfJePvUnbPXyG1NfVzpIwSSDkhqYygg7ZW1gcxO2XH6zlvlxva840ofIZB0QFIbQwGp\nPSHbG1n2zOKl3c/sHtjZ/e3Z78nq6fnHlT5CIOmApDaGAlJ2YLWMb1onI092/+YKua/z19+X\nVRN5h4A0PyB5N4YDUnZ059olq254LTsNaUJO9WMgzQ9I3o0hgXRulT5CIOmApDaABCQjIHk3\ngAQkIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMg\neTeABCQjIHk3gAQkIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMgeTeABCQjIHk3gAQk\nIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMgeTeA\nBCQjIHk3gAQkIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5\nN4AEJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMgeTeABCQj\nIHk3gAQkIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5N4AE\nJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMgeTeABCQjIHk3\ngAQkIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMg\neTeABCQjIHk3gAQkIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMgeTeABCQjIHk3gAQk\nIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMgeTeA\nBCQjIHk3gAQkIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5\nN4AEJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMgeTeABCQjIHk3gAQkIyB5N4AEJCMgeTeABCQj\nIHk3gAQkIyB5N4AEJCMgeTc+2JD+y65pIM0PSN4rGEi12i9c9iKQdEDyXsFA+vlap1/f/i6Q\nCgHJewUD6cTXfvtjHUo/t+J/tYDUD0jeKxhInQ7v/M8f6Vj659e9CaReQPJewUDq9aNb/32H\n0kf+232zQDoEJP8VDKRT7fvcr3Ys/eNP7wMSkNxXMJAKvfCfut94+OjqHwEJSM4rGEj9Xrv2\nX3YUffg3/0nno9ITQAKS7woGUmjyj/9j94PRL139ejZz+y/U/t4rlSE1y2q1St/dKWvHjmi/\nFxut+Eb5Mw0b8Uc6gI34k43/W7obSVdwbyNp4kO905I4MZjLo/S9jbNDevbif9D9YPRb35gL\nf/v6P62NVYZUSp2PSDo+IqmND/ZHpB/f9Gvhg9E1Pzh119drvwik2AaQ1BUMpI92Pxj91/vn\nCncdrlX/udbSRwgkHZDUxgcbUq32y5vnfcd7qvYPgRTbAJK6goH03x9szMeQNZpAim0ASV3B\nQBpwpY8QSDogqQ0gAckISN4NIAHJCEjeDSAByQhI3g0gAckISN4NIAHJCEjeDSAByQhI3g0g\nAckISN4NIAHJCEjeDSAByQhI3g0gAckISN4NIAHJCEjeDSAByQhI3g0gAckISN4NIAHJCEje\nDSAByQhI3g0gAckISN4NIAHJCEjeDSAByQhI3g0gAckISN4NIAHJCEjeDSAByQhI3g0gAckI\nSN4NIAHJCEjeDSAByQhI3g0gAckISN4NIAHJCEjeDSAByQhI3g0gAckISN4NIAHJCEjeDSAB\nyQhI3g0gAckISN4NIAHJCEjeDSAByQhI3g0gAckISN4NIAHJCEjeDSAByQhI3g0gAckISN4N\nIAHJCEjeDSAByQhI3g0gAckISN4NIAHJCEjeDSAByQhI3g0gAckISN4NIAHJCEjeDSAByQhI\n3g0gAckISN4NIAHJCEjeDSAByQhI3g0gAckISN4NIAHJCEjeDSAByQhI3g0gAckISN4NIAHJ\nCEjeDSAByQhI3g0gAckISN4NIAHJCEjeDSAByQhI3g0gAckISN4NIAHJCEjeDSAByQhI3g0g\nAckISN4NIAHJCEjeDSAByQhI3g0gAckISN4NIAHJCEjeDSAByQhI3g0gAckISN4NIAHJCEje\nDSAByQhI3g0gAckISN4NIAHJCEjeDSAByQhI3g0gAckISN4NIAHJCEjeDSAByQhI3g0gAckI\nSN4NIAHJCEjeDSAByQhI3g0gAckISN4NIAHJCEjeDSAByQhI3g0gAckISN4NIAHJCEjeDSAB\nyQhI3g0gAckISN4NIAHJCEjeDSAByQhI3g0gAckISN4NIAHJCEjeDSAByQhI3g0gAckISN4N\nIAHJCEjejUFBSt0AUghIOiB5N4AUApIOSN4NIIWApAOSdwNIISDpgOTdAFIISDogeTeAFAKS\nDkjeDSCFgKQDkncDSCEg6YDk3fgZQzq4Y+2S8a179Z2t+0blbSBZAcm7MRyQXl8pq7ZsFHmo\neOdPrhEBkh2QvBtDAam5Qe5sZtlLyxbvO33ncytGH1gKJDsgeTeGAtIeWd/u3u6SW07f+YWL\nX82AdJaA5N0YCki3yr3h9oCMtU7d+djJDEhnC0jejaGAtFHybzOs0HCAdLaA5N0YCkjL5K3e\nG5fL80A6BKR5Vx+Qzq2myKHeW1fLU2eB9NVt27b98UxZjUbpuztlrdgRzXrkgNmsGdtozUYO\nqGfRR9qOHTCXzSVvNLLos42/YFn02XZfsKSrr/NsG4kTnY3GXPLDCJdH6kbk8kiCNC1yrPfW\nFnn8LJDWLFq0aHWFbTofSrr6BjAxiI3BPZWSTn+LoAqk1jl8RHrjlVdeeeNoWdPTpe/u1G7G\njqhPRg44ns3FNuaORw44mc3GNlrHIgdMZdFn24odMJNFn20jtlHPos+23vlL0tXXebYziROd\njZmp5IdxtH4yfaNZ+oIdT4KULZf9vTcukxfOAilU+sk4XyPp+BpJbQzD10jZlbmf9pi8A6RD\nQJp39QHpHLtddoXbfTLeBtIhIM27+oB0ju2Vtc3u7U7Zod8BpLMEJO/GUEBqT8j2RpY9s3hp\n9zO7B3ae+ilwIJ0lIHk3hgJSdmC1jG9aJyNPdv/mCrmv89d3JjqNyMbOX0993VR6xoCkA5La\nGA5I2dGda5esuuG17DSkN+VUbwJpfkDybgwJpHOr9IwBSQcktQEkIBkBybsBJCAZAcm7ASQg\nGQHJuwEkIBkBybsBJCAZAcm7ASQgGQHJuwEkIBkBybsBJCAZAcm7ASQgGQHJuwEkIBkBybsB\nJCAZAcm7ASQgGQHJuwEkIBkBybsBJCAZAcm7ASQgGQHJuwEkIBkBybsBJCAZAcm7ASQgGQHJ\nuwEkIBm9h5ASrxwg6Q0ghYDkvXKApDeAFAKS98oBkt4AUghI3isHSHoDSCEgea8cIOkNIIWA\n5L1ygKQ3gBQCkvfKAZLeAFIISN4rB0h6A0ghIHmvHCDpDSCFgOS9coCkN4AUApL3ygGS3gBS\nCEjeKwdIegNIISB5rxwg6Q0ghYDkvXKApDeAFAKS98oBkt4AUghI3isHSHoDSCEgea8cIOkN\nIIWA5L1ygKQ3gBQCkvfKAZLeAFIISN4rB0h6A0ghIHmvHCDpDSCFgOS9coCkN4AUApL3ygGS\n3gBSCEjeKwdIegNIISB5rxwg6Q0ghYDkvXKApDeAFAKS98oBkt4AUghI3isHSHoDSCEgea8c\nIOkNIIWA5L1ygKQ3gBQCkvfKAZLeAFIISN4rB0h6A0ghIHmvHCDpDSCFgOS9coCkN4AUApL3\nygGS3gBSCEjeKwdIegNIISB5rxwg6Q0ghYDkvXKApDeAFAKS98oBkt4AUghI3isHSHoDSCEg\nea8cIOkNIIWA5L1ygKQ3gBQCkvfKAZLeAFIISN4rB0h6A0ghIHmvHCDpDSCFgOS9coCkN4AU\nApL3ygGS3gBSCEjeKwdIegNIISB5rxwg6Q0ghYDkvXKApDeAFAKS98oBkt4AUghI3isHSHoD\nSCEgea8cIOkNIIWA5L1ygKQ3gBQCkvfKAZLeAFIISN4rB0h6A0ghIHmvHCDpDSCFgOS9coCk\nN4AUApL3ygGS3gBSCEjeKwdIegNIISB5rxwg6Q0ghYDkvXKApDeAFAKS98oBkt4AUghI3isH\nSHoDSCEgea8cIOkNIIWA5L1ygKQ3gBQCkvfKAZLeAFIISN4rB0h6A0ghIHmvHCDpDSCFgOS9\ncoCkN4AUApL3ygGS3gBSCEjeKwdIegNIISB5rxwg6Q0ghYDkvXKApDeAFAKS98oBkt4AUghI\n3isHSHoDSCEgea8cIOkNIIWA5L1ygKQ3gBQCkvfKAZLeAFIISN4rB0h6A0ghIHmvHCDpDSCF\ngOS9coCkN4AUApL3ygGS3gBSCEjeKwdIegNIISB5rxwg6Q0ghYDkvXKApDfOC0gny6rPlr67\nU9aKHTE3EzlgMmvENppTkQOms7nYRvSBzmTRZ9uOHVDPpiNHTDY7f0m6cjrPtpG+MVtPnPhQ\n7/JInDjZmE7faE6WvuLvDaTJsur10nd3ylqxI+ZmIgdMZY3YRnM6csBMNhfbiD7Q2Sz6bNux\nA+pZ9Nk2O39JunI6z7aRvjE7lzjR2ZibTX4Yk42Z9I3WVOkr/t5AKv08hE/tdHxqpzb41A5I\nRkDybgAJSEZA8m4ACUhGQPJuAAlIRkDybgAJSEZA8m4ACUhGQPJuAAlIRkDybgAJSEZA8m4A\nCUhGQPJuAAlIRkDybgAJSEZA8m4ACUhGQPJuAAlIRkDybgAJSEZA8m4ACUhGQPJuAAlIRkDy\nbgAJSEZA8m4ACUhGQPJuAAlIRkDybgAJSEZA8m4ACUhGQPJuAAlIRkDybgAJSEZA8m4ACUhG\nQPJuAAlIRkDybgAJSEZA8m4ACUhGQPJuAAlIRkDybgAJSEZA8m4ACUhGQPJuAAlIRkDybgAJ\nSEZA8m4ACUhGQPJuAAlIRkDybgAJSEZA8m4ACUhGQPJuAAlIRkDybgAJSEZA8m4ACUhGQPJu\nAAlIRkDybgAJSEZA8m4ACUhGQPJuAAlIRkDybgAJSEZA8m4ACUhGQPJuAAlIRkDybgAJSEZA\n8m4ACUhGQPJuAAlIRkDybgAJSEZA8m4ACUhGQPJuAAlIRkDybgAJSEbnCCnxrANJTQApD0jO\nsw4kNQGkPCA5zzqQ1ASQ8oDkPOtAUhNAygOS86wDSU0AKQ9IzrMOJDUBpDwgOc86kNQEkPKA\n5DzrQFITQMoDkvOsA0lNACkPSM6zDiQ1AaQ8IDnPOpDUBJDygOQ860BSE0DKA5LzrANJTQAp\nD0jOsw4kNQGkPCA5zzqQ1ASQ8oDkPOtAUhNAygOS86wDSU0AKQ9IzrMOJDUBpDwgOc86kNQE\nkPKA5DzrQFITQMoDkvOsA0lNACkPSM6zDiQ1AaQ8IDnPOpDUBJDygOQ860BSE0DKA5LzrANJ\nTQApD0jOsw4kNQGkPCA5zzqQ1ASQ8oDkPOtAUhNAygOS86wDSU0AKQ9IzrMOJDUBpDwgOc86\nkNQEkPKA5DzrQFITQMoDkvOsA0lNACkPSM6zDiQ1AaQ8IDnPOpDUBJDygOQ860BSE0DKA5Lz\nrANJTQApD0jOsw4kNQGkPCA5zzqQ1ASQ8oDkPOtAUhNAygOS86wDSU0AKQ9IzrMOJDUBpDwg\nOc86kNQEkPKA5DzrQFITQMoDkvOsA0lNACkPSM6zDiQ1AaQ8IDnPOpDUBJDygOQ860BSE0DK\n+2BBSj1jQFIbQAJStTMGJLUBJCBVO2NAUhtAAlK1MwYktTEkkA7uWLtkfOveyH2lFx+Q9BkD\nktoYDkivr5RVWzaKPFR+X+nFByR9xoCkNoYCUnOD3NnMspeWLd5Xel/pxQckfcaApDaGAtIe\nWd/u3u6SW0rvK734gKTPGJDUxlBAulXuDbcHZKxVdl/pxQckfcaApDaGAtJGyb+lsELeLruv\n9OIDkj5jQFIbQwFpmbzVe+Nyeb7svtKLD0j6jAFJbQwDpKbIod5bV8tTZ7nvpwcOHDh4pKyp\nqdJ3d2o3Y0fMnowccCybi23MHTtyJOnV7mw0j6ZvtI4kbxxtpG+cqKdvTE0nTnQ2pqfSX9L6\nicGc2rLrKwXStEg+sEUeP8t9axYtWrS6wvb7U9KrzYa1kTQxiI0Bvhxn79S3AypBahkfkebf\nt2PTpk13zJbVaJS+u1PWjh3RnIscUM9asY1WPXLAXNaMbbRjG40s+myjT7aRRZ9t9Mk2s+iz\njb9gWeyIRvwFi5/a+OURf6TxyyP11CZBypbL/t4bl8kLZfeVfl2x0L5GKu0c/zOK0qay6LOd\ni23MZtFnOxvbmCs/r4d6XyOV14o/29gLdjiLP9tjkQOOZDOxjZ/pNxuuzK20x+SdsvtKHyGQ\ndEBSDQWk22VXuN0n4+2y+0ofIZB0QFINBaS9srbZvd0pO0rvK32EQNIBSTUUkNoTsr2RZc8s\nXtr9LO6BnXvPuA9IOiDpgNTrwGoZ37RORp7s/r2up7UAAAlHSURBVM0Vct8Z9wFJByQdkPKO\n7ly7ZNUNr2UFSOo+IOmApAOSq9JHCCQdkFRAApIRkHRAApIVkFRAApIRkHRAKg4AqR+QVEDS\nASkEJB2QVEDKA5IKSDog9QKSCkgqIOUBSQckFZCAZAQkHZCAZAUkFZCAZAQkHZCKA0DqByQV\nkHRACgFJByQVkPKApAKSDki9gKQCkgpIeUDSAUkFJCAZAUkHJCBZAUkFJCAZAUkHpOIAkPoB\nSQUkHZBCQNIBSQWkPCCpgKQDUi8gqYCkAlIekHRAUgEJSEZA0gHpveyLX0meOLHtweSNt7Y9\nkbzx8rYXkzee3vaD5I1vb3s3eePuba34QZH+aHvyRHPb15M3Dm/7dvJG3vkM6T98PHnip4uu\nTt743qI/TN54ZNE3kjfuXPRc8sbWRW8mb6xd1Eje+O3/kTxRX7QueeONRdclb+QBKRqQdECy\nAlI0IOmAZAWkaEDSAcnqfIZE9IEJSEQDCEhEAwhIRAPo/IXUum9U3k6b2H/bmtELrvpWM2Xj\n5RtWj6645s+SNrq9MiLPVv+n56Rf0m8O7/m9j49dfv909YFrTj0O+XHlke/ftHp05ebvJL2k\nr/3BRaOrbqr8O9TFi+vgjrVLxrfuTXk02XkM6Sfdc5YG6ZklsmLiEpHNc9U3dot84qrOxqZ6\n0kPJ6pdKCqQjMjrR67sJD+IuGbny6qXyqZnKC9vzRzExIj+tuvHwiFx7x01jsiXhJX10RNZs\nuVhG91T7x4sX1+srZdWWjSIPVX803c5XSM+tGH1gaRqkQ8vly51z9dJyebjyxl/L4ifanY0x\nSfxRo6/IhSmQ3pZL0v793Z6Wizq/hB/eKLuTp56Vm6r+owcWj3R/XOrQJXJ/5X/9/lHZ1c7a\nu2TsWJV/vHhxNTfInZ2PjS8tW7yv8sPpdr5C+sLFr2aJkHbLRLt7e7dcU3njZtmZ9TY2pTyU\n7NWRzTemQHpNPpX07w9dKt8Lj2VnwgPpNXXR2KGq/+yDcm24/ZZ8uvK/f6dsCbdbZVeVf7x4\nce2R9eEq2SW3VH443c5XSI+dzFIh7b3nr8Ltc7K28sb+Fw+G27+QiZSHUl+/7EASpL2yOeXf\nH/p/sjF5I29nwidCd8tt4fZZubTyxoQ8Em6fr/brS/HiulXuDbcHZCzpZ3HPV0jdEiH1+07C\nR6R+X5Y7Uv7xr8g3syRIT8vnU/79oYfky8kbvX44cln17xQ8kZ+Oh+RzlTc+Kc+H2wMyWvWB\n9C+ujZJ/m2FF2tW28CG1PyOPJk5M3jcyfjDhn3915KpWGqTH5JYf3HX99Xe9lvAo/lAefftL\nn1iy7q4TCSOhbfJ/qv/DM6vlLzs3x9ZK9f+yZEIeC7cHRd6pONG/uJbJW707Ls9xVmzhQ7pf\nNiZ9o/Xw+jWjF3yp8pcEWfcTuyWdk5UE6ZtyYe+bzne0K29slT8Z+53rty6XdZW+Qj/dD0eS\nPs9941L57B03rViZ8Mvb/5St4fbbIj+sOJFfXE2R/NReLU9Vf0BDAOmBkQur/qLVq/Ornqz8\n/MsJC3eF708lQbpblu76ceOnXx2VeypvbJLF93R+SfnJ5YlfVmc3yNMp/3j94RXdXxJue6P6\nxL6R8J3Y55cvkdcrTuQX17RI/svKFnm8+gNa8JCat8tFyT/vXN+/e/lI9Wvn1ZGJ7ofEJEhH\n3uz9l63flOWVfxPos/K74fZvZDTht2Q7EkdWpXyIn/qU/NGPmsefvHBpwqdSu0XWf/4y+dqF\ncqDiQn5xtfiIdC5NXiuXp3xxc6o9sqrq7+p2PrF7o3ubBOnU2GKp/LHx+vybDe1l8mrKY9id\n9k2Lr8gXw+0LclHC78g+d83Yis3PNkdlsuJA/+JaLvt7d1wmL1R/OAsc0uQVcl3SL76nu0Cq\n/jjKo7Lmxm6/I5+5Mf1PCLhEKv5mfvc7j/kflHCx/N+Uh7CxuuVul0jvdyU6nv82Zafbm7Km\n6j/av7iuzP20xyp/3yK0kCHNXiO3pf2MXOvmzb3v6XTOetWr56HTP55W/YuT9ru9J9Iak+9X\n3fjf0vuPHDu/jKf8Jv4hWZbwI1dZNt5nfIGk/nxb57Wt/Ir2L67b89/T3Sfj1b+Pky1sSLfK\njUmvTdb9PuvXwu2rIocTp5I+tbtUngy3T8qyyp8PTS7tAdojF6T88vJ02u9Nd17S3n8s/E7C\nz73+zU13d2/qa6tb7F9ce2VteDV2yo6qU6EFDOl7sjr587pHZPTPOxjf2JD/XEtCSZDukY+/\n1Ll5YVz+pPrIXbK+89nLDy9J2ciyryd+0+9PZbz7nbbGF+Sqyhv7R0Y7gmZulM9WnuhfXO0J\n2d7IsmcWL0375u55Cumd3k8Yb+z8tfLz+31Z1f9Z5cq/C9S+RWT80+tG5JLk71kkQapvEbn4\n6otErk/48xI6I0s/e/WobE76SfZbJe2PG2zeIIuv3f4Hq2V11W+4dbpP5IrfWymfPBI/9Mz0\nxXVgtYxvWicjT1Z/NN3OU0hvnv66ovJ3rycG8d/OZHs+t2r0git3V/3m0OnSvmvXemzzytHx\na59K+ly1+a0rx5Zf+XDSlzjZ9dV+UPR07ae3dl/Se5J+wOKZTePL1n+92mcc8y6uozvXLll1\nQ8rPjHQ7TyERfbACEtEAAhLRAAIS0QACEtEAAhLRAAIS0QACEtEAAhLRAAIS0QACEtEAAtKC\naP8/qv1678dZJ3+l9kuV/zRhqhyQFkZ312q9P0R4olb70/f5sQxlQFogLav93e5/DP/iR2vV\n/1xZqh6QFkiHfrn2W1nW/He1f3Hy/X4oQxmQFkqP1GpfzW6ufWQAf1YR+QPSgumTtV/87t+v\npf9h+1QlIC2YJn+19nO1f5v4P0SjigFp4fSXtVrtpff7QQxrQFo4XduBdP37/SCGNSAtmL77\nd2q/UftY8h+6SJUC0kJp+tdqvzn7b2r/uvr/aZkSAtJC6VO1n38te+Gjtd99vx/IcAakBdKT\nH651/y8Pn6l9OPFPOqRKAWlhdPxXar/R/TOsp/9V7Z8l/g/5qEpAWhitrn2s97/LeOrDtU+8\nz49lKAPSgujhWu26/M31tdqD7+tjGc6ARDSAgEQ0gIBENICARDSAgEQ0gIBENICARDSAgEQ0\ngIBENICARDSAgEQ0gIBENICARDSAgEQ0gIBENICARDSA/j8WQ7v+2E3w/QAAAABJRU5ErkJg\ngg==" }, "metadata": { "image/png": { "width": 420, "height": 420 } } } ] }, { "cell_type": "markdown", "source": [ "また、`pbinom`関数を使えば、二項分布の累積分布関数を計算出来ます。\n", "\n", "その事象が$x$回以下生じる確率の合計を求めます。\n", "\n", "```\n", "pbinom(x, size=試行回数, prob=事象の確率)\n", "```\n", "\n", "80%の確率で勝つ勝負を10回繰り返したときに、多くとも5回しか勝つ事が出来ない確率は..." ], "metadata": { "id": "TpZwbos7UUzd" } }, { "cell_type": "code", "source": [ "# n=10, p=0.8の二項分布でxが5以下の確率を求める\n", "pbinom(5, 10, 0.8)" ], "metadata": { "id": "clVw7qpgrTTe", "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "outputId": "0d17f65b-be54-4f41-c737-66bf2341a8d3" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "0.1208738816" ], "text/markdown": "0.1208738816", "text/latex": "0.1208738816", "text/plain": [ "[1] 0.1208739" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "`pbinom`関数を使用すれば、80%の確率で勝つ勝負を10回繰り返したときに、5回または6回勝つ確率、等も求められます。" ], "metadata": { "id": "y64FX3euY50j" } }, { "cell_type": "code", "source": [ "# n=10, p=0.8の二項分布でxが5か6の確率を求める\n", "pbinom(6, 10, 0.8) - pbinom(4, 10, 0.8)" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "id": "E6U7zESBZD6x", "outputId": "68ebe0b9-d1a4-4d3a-d82f-71698adbde39" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "0.1145044992" ], "text/markdown": "0.1145044992", "text/latex": "0.1145044992", "text/plain": [ "[1] 0.1145045" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "正規分布の時と同じように、ランダムサンプリングが出来る`rbinom`関数やパーセンタイル点を求める`qbinom`関数もあります。\n", "\n", "`rbinom`関数は二項分布に従う乱数を$n$個生成します。\n", "\n", "```\n", "rbinom(n, size=試行回数, prob=事象の確率)\n", "```\n", "\n", "`qbinom`関数は、指定した累積確率$p$になる最小の成功回数を求めます。\n", "\n", "```\n", "qbinom(p, size=試行回数, prob=事象の確率)\n", "```" ], "metadata": { "id": "L1GUtPpLXfXW" } }, { "cell_type": "code", "source": [ "# n=10, p=0.8の二項分布から無作為に7回抽出する\n", "rbinom(7, 10, 0.8)" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "id": "JiwLGOssXEFj", "outputId": "75a7c269-38d8-4a17-b072-f2e61131bcd8" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "\n", "
  1. 9
  2. 10
  3. 8
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\n" ], "text/markdown": "1. 9\n2. 10\n3. 8\n4. 8\n5. 9\n6. 9\n7. 9\n\n\n", "text/latex": "\\begin{enumerate*}\n\\item 9\n\\item 10\n\\item 8\n\\item 8\n\\item 9\n\\item 9\n\\item 9\n\\end{enumerate*}\n", "text/plain": [ "[1] 9 10 8 8 9 9 9" ] }, "metadata": {} } ] }, { "cell_type": "code", "source": [ "# n=10, p=0.8の二項分布で、累積確率が0.5になる回数を求める\n", "qbinom(0.5, 10, 0.8)" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "id": "8rf1dTPsYFu5", "outputId": "2459348e-8e43-42f2-cbab-523bcc75262a" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "8" ], "text/markdown": "8", "text/latex": "8", "text/plain": [ "[1] 8" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "この様に、各確率分布毎にRの関数が用意されています。例: `dnorm`, `dbinom`\n", "\n", "それぞれの確率分布の形を決めるパラメータを指定することで関数が使用できます。\n", "\n", "確率分布の形は\n", "\n", "* 正規分布$N(\\mu, \\sigma)$の場合は、平均値と標準偏差。\n", "\n", "* 二項分布$Bi(n, p)$の場合は、試行回数と事象の確率\n", "\n", "によって形が決まります。" ], "metadata": { "id": "jkImTs8Wrkg-" } }, { "cell_type": "markdown", "source": [ "### ポアソン分布\n", "\n", "二項分布において、$n$が大きく(大量に観察を行い)、$p$が小である(極めて珍しい現象がある)場合を考えると、\n", "\n", "現象自体は珍しいものの観察数が多いので、ある程度$x$が観察される場合があります。\n", "\n", "例えば、交通事故などが例に挙げられます。(交通事故自体が起きる確率は稀だが、交通量が多いためそれなりに事象としては観察される。)\n", "\n", "外出した際に交通事故にあう確率が$p = 0.0001$(0.01%)として、10000回(約27年間毎日)外出した際に交通事故に１度あう確率を計算してみると、その確率は二項分布の式から、\n", "\n", "$f(1) = _{10000}C_1(0.0001)^1(0.9999)^{9999}$\n", "\n", "で計算できます。`dbinom`関数で計算してみると…\n", "\n" ], "metadata": { "id": "DiAbQF7jANCe" } }, { "cell_type": "code", "source": [ "# n=10000, p=0.0001の二項分布でx=1となる確率を求める\n", "dbinom(1, size=10000, p=0.0001)" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "id": "WVtcBsYVaCEx", "outputId": "58ca8682-2e34-47a6-d6ec-70ec189f7045" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "0.367897836216552" ], "text/markdown": "0.367897836216552", "text/latex": "0.367897836216552", "text/plain": [ "[1] 0.3678978" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "と、それなりの確率で起きる事象だと分かります。\n", "\n", "二項分布の期待値$E[X]=np=10000\\times0.0001=1$というところからも、$x=1$の確率はそれほど低くないと考えられます。\n", "\n", "この様な、膨大な観察数を行うことで得られる、極めて起きる確率が低い事象の起きる数である確率変数$X$は、\n", "\n", "二項分布にしたがうだけでなく、**ポアソン分布**と呼ばれる確率分布に限りなく近づくことが知られています。\n", "\n", "詳細な計算は省きますが、$np\\to\\lambda$となるように$n\\to\\infty, \\space p\\to0$となる極限においては、\n", "\n", "$_nC_xp^x(1-p)^{n-x}\\to e^{-\\lambda}\\dfrac{\\lambda^x}{x!}$\n", "\n", "が成り立つことがポアソンの小数の法則として知られています。ここで、\n", "\n", "$f(x) = e^{-\\lambda}\\dfrac{\\lambda^x}{x!}$という確率関数に従う分布を**ポアソン分布Poisson distribution**と呼び、$Po(\\lambda)$で表します。\n", "\n", "ポアソン分布$Po(\\lambda)$に従う確率変数$X$の期待値と分散は、\n", "\n", "$E(X) = \\lambda$\n", "\n", "$V(X) = \\lambda$\n", "\n", "となり、期待値と分散が等しくなることがポアソン分布の特徴です。" ], "metadata": { "id": "OoRQzgluZ7XU" } }, { "cell_type": "markdown", "source": [ "ポアソン分布に関係するRの関数は`dpois`, `ppois`, `rpois`, `qpois`になります。\n", "\n", "ポアソン分布$Po(\\lambda)$の形を決めるパラメータは$\\lambda = np$だけになります。\n", "\n", "例えば先ほどの交通事故にあう確率が$p = 0.0001$として、10000回外出した場合を考えると、\n", "\n", "$np = 0.0001 \\times 10000 = 1$\n", "\n", "となるので、交通事故にあう回数は$Po(1)$にある程度従うと考えられます。\n", "\n", "ポアソン分布に従う事象が$x$回起きる確率は`dpois`関数で計算できます。\n", "\n", "```\n", "dpois(x, lambda=ラムダ値)\n", "```\n", "\n", "起きる回数が$Po(1)$に従う事象が1度起きる確率を計算すると…" ], "metadata": { "id": "BxOsq_Dsrg4O" } }, { "cell_type": "code", "source": [ "# ラムダ=1のポアソン分布に従う事象が1回生じる確率\n", "dpois(1, lambda=1)" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "id": "78cTrORBum8k", "outputId": "032c97c7-db84-4498-acce-8502f4bb0ee6" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "0.367879441171442" ], "text/markdown": "0.367879441171442", "text/latex": "0.367879441171442", "text/plain": [ "[1] 0.3678794" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "と二項分布で計算した場合と非常に近い値となりました。\n", "\n", "$x=0 \\sim 10の範囲で$ポアソン分布$Po(1)$のグラフを描いてみると下図のようになります。" ], "metadata": { "id": "0ujKaiMvvDcY" } }, { "cell_type": "code", "source": [ "# ポアソン分布の描写\n", "library(ggplot2)\n", "x <- seq(0,10,1) # x = 0 ~ 10の範囲を描写\n", "m <- dpois(x,lambda=1) # ラムダ=1のポアソン分布\n", "data <- data.frame(x=x,y=m)\n", "g <- ggplot(data,aes(x=x, y=y))\n", "g <- g + geom_bar(stat = \"identity\", position = \"dodge\", fill = \"#00A968\") # 棒グラフを追加\n", "g <- g + theme(text = element_text(size = 18)) # フォントサイズ変更\n", "g <- g + scale_x_continuous(breaks = seq(0,10,1)) # 軸ラベル追加\n", "g" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 437 }, "id": "3qsfrSuuvbUf", "outputId": "c080b4cf-a54e-48a7-a9b5-179b5f9d078e" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/plain": [ "plot without title" ], "image/png": 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0giINkdgCRmA0kEJLsDkMRsIImAZHcA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\nRU5ErkJggg==" }, "metadata": { "image/png": { "width": 420, "height": 420 } } } ] }, { "cell_type": "markdown", "source": [ "これまで同様、1回から6回のいずれかの回数この事象が起きる確率を`ppois`関数で計算してみると" ], "metadata": { "id": "CZ5ygVB-wFed" } }, { "cell_type": "code", "source": [ "# ラムダ=1のポアソン分布に従う事象が1~6回のいずれか生じる確率\n", "ppois(6, lambda=1) - ppois(0, lambda=1)" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "id": "YVHQU4wLwRyK", "outputId": "db27ebf4-08a6-47ad-bd49-4b57acda632b" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "0.63203731767927" ], "text/markdown": "0.63203731767927", "text/latex": "0.63203731767927", "text/plain": [ "[1] 0.6320373" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "それなりの確率で交通事故に遭うかもしれないということが分かります。" ], "metadata": { "id": "pXgT_txIwmR5" } }, { "cell_type": "markdown", "source": [ "#### 二項分布とポアソン分布の関係\n", "\n", "さて、ここまでRの関数を使用できたので、ややこしい計算をする必要がありませんでした。\n", "\n", "そのため、二項分布をポアソン分布で近似する必要性を感じられなかったかもしれないですが、\n", "\n", "実際に計算することを考えると、この近似はとても便利なことが分かります。\n", "\n", "\"title\"\n", "\n", "例えば、ある地域では0.2%の確率で四葉のクローバーが出現するとします。\n", "\n", "四葉のクローバーを求めてシロツメクサを500個体採集した際に、四葉のクローバーが2個体採集できる可能性を考えてみます。\n", "\n", "二項分布でまず考えてみると、この条件で観察される四葉のクローバーの数である確率変数$X$は$p=0.002, n=500$の二項分布に従います。\n", "\n", "$X \\sim Bi(500, 0.002)$\n", "\n", "この時、採集した500個体の中に2個体四葉がある確率は\n", "\n", "$f(x=2)=_{500}C_2(0.002)^2(0.998)^{498}$となります。\n", "\n", "関数を使わず素直にRで計算してみると" ], "metadata": { "id": "GBpPp-k2ijnT" } }, { "cell_type": "code", "source": [ "# 500個体中0.2%の個体が2個体いる確率\n", "500*499/2*(0.002^2)*(0.998^498)" ], "metadata": { "id": "rT8MPPgckFAa", "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "outputId": "19b8e8e4-8344-40ff-87b4-b2fac3b364f4" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "0.184123875178937" ], "text/markdown": "0.184123875178937", "text/latex": "0.184123875178937", "text/plain": [ "[1] 0.1841239" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "続いて、ポアソン分布を用いて考えてみます。見つかる確率$p$は小さく、観察数$n$は大きいので、この確率変数の確率分布は\n", "\n", "パラメータが、$\\lambda = np = 500 \\times 0.002 = 1$であるポアソン分布に近似できると考えられます。\n", "\n", "確率変数$X$は$\\lambda=1$のポアソン分布$Po(1)$に従うので、\n", "\n", "$f(x) = e^{-1}\\dfrac{1^x}{x!}$に従う形になります。\n", "\n", "この場合、500個体中2個体が四葉である確率は\n", "\n", "$f(x=2) = e^{-1}\\dfrac{1^2}{2!}$となります。\n", "\n", "こちらも関数を使わず素直にRで計算してみると" ], "metadata": { "id": "PAi88_R5k1j_" } }, { "cell_type": "code", "source": [ "# ポアソン分布Po(1)に従う個体数が2個体の確率を求める\n", "exp(-1) * 1 / 2" ], "metadata": { "id": "tzqRtzbIl_xX", "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "outputId": "e7605bcc-648e-4eca-fef0-505deaee5c07" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "0.183939720585721" ], "text/markdown": "0.183939720585721", "text/latex": "0.183939720585721", "text/plain": [ "[1] 0.1839397" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "といった形で、二項分布より簡単な計算でほぼ等しい確率が計算できました。\n", "\n", "ポアソン分布を仮定できる現象は非常に多く、上述した交通事故や珍しい形質を持つ個体の数だけでなく、\n", "\n", "遺伝子の突然変異数や顕微鏡視野に入るバクテリア数など、様々な生物現象に仮定としておくことができます。" ], "metadata": { "id": "6xcXkPmQnatV" } }, { "cell_type": "markdown", "source": [ "### そのほかの分布\n", "\n", "代表的な分布として、正規分布、二項分布、ポアソン分布を紹介しました。\n", "\n", "他にも負の二項分布、一様分布、$\\chi^2$分布、指数分布、ガンマ分布など様々な分布が存在しています。\n", "\n", "また、二項分布とポアソン分布の関係性のように、分布同士でつながりの強いものも多いです。\n", "\n", "確率論を勉強していくと一通り学ぶことになりますし、こういった↓まとめも探すと見つけることが出来るかと思います。\n", "https://www.math.wm.edu/~leemis/chart/UDR/UDR.html\n", "\n", "本講義では全て取り上げるとキリがないのでこのあたりにしておきますが、\n", "\n", "各種統計手法の理解のために必要な分布($t$分布や$F$分布、$\\chi^2$分布など)については、その項目を扱う際に触れたいと思います。" ], "metadata": { "id": "SM45ctS7oag8" } }, { "cell_type": "markdown", "source": [ "## 大数の法則と中心極限定理\n", "\n", "最後に、確率論の定理の中でも、統計学の上で大きな意味合いを持つ**大数の法則**と**中心極限定理**について触れておきます。\n" ], "metadata": { "id": "e4nWTJGwbVxm" } }, { "cell_type": "markdown", "source": [ "### 大数の法則\n", "\n", "大数の法則とは、**試行回数が増えるにつれて、実際に得られる確率が理論上の確率に近づく**という法則を指します。\n", "\n", "当たり前の様な気がしますが、実際の例でみてみましょう。\n", "\n", "例えば、先ほどの生息しているシロツメクサの20%が薄紅色で80%が白色の地域において、10個体標本を採集した時に観察される薄紅色のシロツメクサの数を考えてみます。\n", "\n", "薄紅色の個体数$X$は二項分布$Bi(10, 0.2)$に従うので、薄紅色の個体が0,1,2,...9,10個体観察される確率をそれぞれ計算してグラフ化してみると" ], "metadata": { "id": "WZ_A0VQkyi0C" } }, { "cell_type": "code", "source": [ "# 観察数10, 事象の確率0,2の二項分布\n", "n = 10\n", "p = 0.2\n", "x <- seq(0, n, 1) # 0 ~ 10個体\n", "m <- dbinom(x, size = n, prob = p) # n=10, p=0.2の二項分布における確率\n", "data <- data.frame(x=x,y=m)\n", "g <- ggplot(data,aes(x=x, y=y))\n", "g <- g + geom_bar(stat = \"identity\", position = \"dodge\", fill = \"#00A968\") # 棒グラフを追加\n", "g <- g + theme(text = element_text(size = 18)) # フォントサイズ変更\n", "g <- g + scale_x_continuous(breaks = seq(0,10,1)) # 軸ラベル追加\n", "g <- g + geom_vline(xintercept = 10*0.2) # 理論上観察される値(n x p)に縦線\n", "g\n" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 437 }, "id": "ajy7UHoh21Zp", "outputId": "19b157d3-64da-4a6d-9734-5beda627a353" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/plain": [ "plot without title" ], "image/png": 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}, "metadata": { "image/png": { "width": 420, "height": 420 } } } ] }, { "cell_type": "markdown", "source": [ "となり、実際の確率(薄紅色が20%)通りに個体が出た場合の、10個体中2個体得られる確率は30%ほどしかないことがわかります。\n", "\n", "ここで採集する個体の数を、100個体、1000個体、、、と増やしていくとどうなるか見ていきます。" ], "metadata": { "id": "h0jMO-sAeKga" } }, { "cell_type": "code", "source": [ "# 観察数n, 事象の確率0,2の二項分布のnを増やしていくとどうなるか\n", "n = 100\n", "p = 0.2\n", "x <- seq(0, n, 1) # 0 ~ n個体\n", "m <- dbinom(x, size = n, prob = p) # n, p=0.2の二項分布における確率\n", "data <- data.frame(x=x,y=m)\n", "g <- ggplot(data,aes(x=x, y=y))\n", "g <- g + geom_bar(stat = \"identity\", position = \"dodge\", fill = \"#00A968\") # 棒グラフを追加\n", "g <- g + theme(text = element_text(size = 18)) # フォントサイズ変更\n", "g <- g + geom_vline(xintercept = n*0.2) # 理論上観察される値(n x p)に縦線\n", "g\n" ], "metadata": { "id": "W085U_kjc2ox", "colab": { "base_uri": "https://localhost:8080/", "height": 437 }, "outputId": "47b7e32d-bf83-47e7-a22d-1a00643e908a" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/plain": [ "plot without title" ], "image/png": 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}, "metadata": { "image/png": { "width": 420, "height": 420 } } } ] }, { "cell_type": "markdown", "source": [ "この様に、観測される結果が理論上の確率付近に集中していきます。これを**大数の法則**と呼びます。\n", "\n", "注意点としては、先ほどの例でいうと1000個体採集した場合も理論値である$1000 \\times 0.2 = 200$に収束しているわけではないので、採集数が増えれば理論上の観測値に収束するというわけではなく、**近傍にある確率が**極めて高くなることを意味します。\n", "\n", "この法則は、**十分な大きさの標本集団を調べれば、母集団の様々な特性をかなり正確に知ることができる**ということを示しており、**統計的推測**において非常に重要な定理になります。" ], "metadata": { "id": "RiGa9gXchHiW" } }, { "cell_type": "markdown", "source": [ "### 中心極限定理\n", "\n", "中心極限定理とは、**母集団がどのような分布であっても、標本サイズ$n$が大きくなると、和$X_1+X_2+X_3+...+X_n$(or平均とする場合もある)の分布は正規分布に近似できる**、という定理になります。\n", "\n", "実際に先ほどの二項分布に従う変数$X \\sim Bi(n, 0.2)$で見てみましょう。\n", "\n", "20%が薄紅色で80%が白色のシロツメクサが生息している地域で標本を\n", "100個体採集して薄紅色の個体が何個体あるかは$Bi(100, 0.2)$に従います。\n", "\n", "これを別の見方で考えると、1個体採集して薄紅色の個体が何個体あるか(薄紅色かどうか)が従う$X \\sim Bi(1, 0.2)$を100回繰り返したものとも考えられます。\n", "\n", "つまり、\n", "\n", "* 1個体目が薄紅色であれば$X_1 = 1$\n", "* 2個体目が白色であれば$X_2 = 0$\n", "* ...\n", "* 100個体目が薄紅色であれば$X_{100} = 1$\n", "\n", "として、得られた観察結果の和$X_1 + X_2 + ... + X_{100}$を$Bi(100, 0.2)$は表しています。\n", "\n", "つまり$Bi(n, 0.2)$の分布は$X \\sim Bi(1, 0.2)$を$n$回繰り返した合計の分布と考える事が出来ます。\n", "\n", "そのため、標本サイズ$n$が大きいほど正規分布に近付けば中心極限定理が成り立っていると考えられます。\n", "\n", "下のコードの$n$の数字を大きくしていくと、和$X_1 + X_2 + ... + X_n$の分布が正規分布に近付くことが分かると思います。" ], "metadata": { "id": "9eCbhWugitxD" } }, { "cell_type": "code", "source": [ "# 中心極限定理、二項分布の場合\n", "n = 5\n", "p = 0.2\n", "x <- seq(0, n, 1) # 0 ~ 10個体\n", "m <- dbinom(x, size = n, prob = p) # n, p=0.2の二項分布\n", "data <- data.frame(x=x,y=m)\n", "g <- ggplot(data,aes(x=x, y=y))\n", "g <- g + geom_bar(stat = \"identity\", position = \"dodge\", fill = \"#00A968\") # 棒グラフを追加\n", "g <- g + theme(text = element_text(size = 18)) # フォントサイズ変更\n", "g\n" ], "metadata": { "id": "7laSkAgJgq5B", "colab": { "base_uri": "https://localhost:8080/", "height": 437 }, "outputId": "30b1e1ee-a567-414c-a6f9-cf8190b41722" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/plain": [ "plot without title" ], "image/png": 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}, "metadata": { "image/png": { "width": 420, "height": 420 } } } ] }, { "cell_type": "markdown", "source": [ "ポアソン分布$Po(\\lambda)$についても見てみます。\n", "\n", "交通事故にあう確率が$p = 0.0001$として、10000回外出した場合何回事故に遭うかは$Po(1)$に従いました。\n", "\n", "そして、$Po(1)$に従う数値をランダム抽出するには\n", "\n", "```\n", "rpois(n, lambda=ラムダ値)\n", "```\n", "\n", "で抽出できました。\n", "\n", "1人調査した場合は`rpois(1, lambda=1)`、10人調査した場合は`rpois(10, lambda=1)`、\n", "\n", "100人調査した場合は`rpois(100, lambda=1)`という形になります。\n", "\n", "\n", "\n" ], "metadata": { "id": "kLvBa72G7GIF" } }, { "cell_type": "code", "source": [ "# ラムダ=1のポアソン分布に従う事象を10人分調査した場合\n", "rpois(10, lambda=1)" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 34 }, "id": "Hpi5G-HEDMZo", "outputId": "8b404252-91ea-4cff-8c71-77a93c480ee7" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "\n", "
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\n" ], "text/markdown": "1. 1\n2. 1\n3. 1\n4. 0\n5. 1\n6. 1\n7. 1\n8. 0\n9. 1\n10. 0\n\n\n", "text/latex": "\\begin{enumerate*}\n\\item 1\n\\item 1\n\\item 1\n\\item 0\n\\item 1\n\\item 1\n\\item 1\n\\item 0\n\\item 1\n\\item 0\n\\end{enumerate*}\n", "text/plain": [ " [1] 1 1 1 0 1 1 1 0 1 0" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "この合計値が調査人数が増えるほど正規分布に近付くというわけです。\n", "\n", "下のコードの$n$の数値を変えて見てみましょう。" ], "metadata": { "id": "1Hlnj5KLDbgJ" } }, { "cell_type": "code", "source": [ "# 中心極限定理、ポアソン分布の場合\n", "n = 1\n", "x <- c()\n", "for (i in 1:1000) {\n", " x <- c(x, sum(rpois(n, 1)))\n", "}\n", "data <- data.frame(x=x)\n", "g <- ggplot(data, aes(x=x))\n", "g <- g + geom_histogram()\n", "g <- g + theme(text = element_text(size = 18))\n", "g\n" ], "metadata": { "colab": { "base_uri": "https://localhost:8080/", "height": 456 }, "id": "YfkhWM5A7Ohz", "outputId": "fc183ae3-fdcf-4532-844c-b0883702a018" }, "execution_count": null, "outputs": [ { "output_type": "stream", "name": "stderr", "text": [ "\u001b[1m\u001b[22m`stat_bin()` using `bins = 30`. Pick better value `binwidth`.\n" ] }, { "output_type": "display_data", "data": { "text/plain": [ "plot without title" ], "image/png": 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KQCkg1ITkBSAckGJCcgqYBkA5ITkFRA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\nB7FFiMA6mtQbAAAAAElFTkSuQmCC" }, "metadata": { "image/png": { "width": 420, "height": 420 } } } ] }, { "cell_type": "markdown", "source": [ "他の分布、例えば下の様な、少し変わった分布に植物の草丈が従う母集団がいたとします。" ], "metadata": { "id": "CyfwzBSHoHTx" } }, { "cell_type": "code", "source": [ "# 二峰型の分布\n", "gene <- function(n){\n", " prob <- (runif(n) <= 0.6)\n", " rlnorm(n,2,0.5) *prob+rnorm(n,30,5)*(1-prob)\n", "}\n", "\n", "height_all <- round(gene(100000))\n", "hist(height_all,xlim=c(0,60),main =\"Plant height\",xlab=\"height(cm)\",ylab=\"samples\",breaks=seq(0,10000,1))" ], "metadata": { "id": "-ZZl-_bin0vZ", "colab": { "base_uri": "https://localhost:8080/", "height": 437 }, "outputId": "100a72ec-12b5-44e9-c773-82e4c8bc8965" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/plain": [ "Plot with title “Plant height”" ], "image/png": 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}, "metadata": { "image/png": { "width": 420, "height": 420 } } } ] }, { "cell_type": "markdown", "source": [ "5cmくらいの植物と、30cmくらいの植物が混在している集団でしょうか。\n", "\n", "この分布に従う植物集団から、$n$個体の標本を採集し、その草丈の合計$X_1+X_2+...+X_n$の分布を考えてみます。\n", "\n", "下のコードは、この分布からランダム抽出した個体の合計値の分布を示しています。\n", "\n", "$n$の部分を増やしていくことで、合計の分布がどうなっていくか見てみましょう。" ], "metadata": { "id": "HgbojWuWuhz6" } }, { "cell_type": "code", "source": [ "# 中心極限定理、二峰型の分布の場合\n", "n <- 1\n", "sum_values <- c()\n", "for(i in 1:1000){\n", " samples <- sample(height_all,n,replace = FALSE)\n", " sum_values <- c(sum_values,sum(samples))\n", "}\n", "hist(sum_values)" ], "metadata": { "id": "by_9ajlgqCXM", "colab": { "base_uri": "https://localhost:8080/", "height": 437 }, "outputId": "1c268631-95a0-4898-da60-f9c4be2a63db" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/plain": [ "Plot with title “Histogram of sum_values”" ], "image/png": 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+CQto1RlRNq5s6ZPr5CjdtumEhIESKksAR/HWnvstHF8ZeRYieuaDHNI6QIEVJY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\ncGizd/5ydWf87A61wnm576BFK66rHPBhh5C2VlfVrbxtSNkz3u0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}, "metadata": { "image/png": { "width": 420, "height": 420 } } } ] }, { "cell_type": "markdown", "source": [ "この様に、母集団がどのような分布であったとしても、標本サイズ$n$が大きくなると、その和(平均)の分布が正規分布に近付いていきます。\n", "\n", "この中心極限定理により、母集団の分布が不明な場合でも、真の平均値が大体どの範囲に収まっているかを推定する「区間推定」や、区間推定の考え方を応用して仮説を検証する「検定」を行うことができるようになります。\n", "\n", "そのため、中心極限定理は統計的推測の強力な基盤となっています。" ], "metadata": { "id": "7H19f9tW3n24" } }, { "cell_type": "markdown", "source": [ "## 標本分布\n", "\n", "では実際に標本抽出によって得られた標本のデータと母集団の関係などについて触れていきます。\n", "\n", "\"title\"\n", "\n", "\n", "最初の方でも触れましたが、これから特徴を知りたい集団全体を**母集団**と呼びました。\n", "\n", "統計的推測では、抽出した標本からこの母数を推定することが1つの目的になります。\n" ], "metadata": { "id": "O9qubXfQ_RMQ" } }, { "cell_type": "markdown", "source": [ "### 確率や分布が必要な理由\n", "\n", "当然ながら、標本を分析して得られる結果は、抽出された標本によって異なる可能性があります。\n", "\n", "仮に、ある公園に生えている植物が5個体(植物1-5)あり、草丈が\n", "\n", "* 植物1: 10cm\n", "* 植物2: 12cm\n", "* 植物3: 14cm\n", "* 植物4: 16cm\n", "* 植物5: 18cm\n", "\n", "であった場合、\n", "\n", "全体から計算された平均値(=母集団の平均値)は14㎝であるのに対し、\n", "\n", "ランダムに3つを抽出した標本をもとに計算される平均値は、植物1,2,3を選ぶと12cm、植物3,4,5を選ぶと16cmと、標本の選び方によって12~16cmと幅が生じてしまいます。\n", "\n", "このような抽出された標本によるばらつきを扱うためには、確率的に標本データを捉える必要があり、**標本分布**という考え方が必要になってきます。\n", "\n", "標本平均の確率分布\n", "* 1/6の確率で植物1,2,3 ... 標本平均12cm\n", "* 1/6の確率で植物1,2,4 ... 標本平均12.666cm\n", "* 1/6の確率で植物1,2,5 ... 標本平均13.333cm\n", "* 1/6の確率で植物2,3,4 ... 標本平均14cm\n", "* 1/6の確率で植物2,3,5 ... 標本平均14.666cm\n", "* 1/6の確率で植物3,4,5 ... 標本平均16cm\n" ], "metadata": { "id": "buhaljMd_cQJ" } }, { "cell_type": "markdown", "source": [ "### 母集団分布\n", "\n", "\"title\"\n", "\n", "統計的推測においては、抽出した標本から母集団の特徴、つまり母集団の従う分布**母集団分布**を知ることが目標となります。\n", "\n", "この母集団分布から標本$X_1, X_2, ... $を抽出してくるので、各$X_i$はこの母集団分布に従う確率変数だと考えることができます。\n", "\n", "### 母数\n", "\n", "母集団が理論的・経験的によく知られた確率分布であることが分かっている場合、\n", "\n", "例えば先ほど扱った交通事故の発生件数は発生確率が非常に低く、母集団分布がポアソン分布だと考えられる。\n", "\n", "つまり、標本$X_1, X_2, ... X_n$は、ある$\\lambda$を持つポアソン分布$Po(\\lambda)$に従うと考えられる。\n", "\n", "このとき、$P(X_i=x)$は$f_\\lambda(x)=e^{-\\lambda}\\dfrac{\\lambda^x}{x!}$となり、定数$\\lambda$さえわかれば母集団について全て知ることができる。\n", "\n", "このような事前に母集団分布=XX分布だと与えられている場合、その分布を決定するパラメータ(今回の場合$\\lambda$)が求めるべきものとなります。\n", "\n", "統計的推測ではこのパラメータを**母数**と呼びます。\n", "\n", "ポアソン分布の場合は$\\lambda$でしたが、\n", "\n", "母集団が正規分布$N(\\mu, \\sigma^2)$に従う場合は、平均$\\mu$と分散$\\sigma^2$の二つの母数によって分布が決定されるので、\n", "\n", "母集団における平均と分散を求めることになります。\n", "\n", "特に母集団分布について事前に分布を決めることができない場合は、母集団の平均(母平均)や中央値、最頻値、母集団の分散(母分散)、レンジなどを考え分析することになります。" ], "metadata": { "id": "9g6pKKXkDOaq" } }, { "cell_type": "markdown", "source": [ "### 統計量\n", "\n", "統計的推測では、抽出した標本からこの母数を推定することが1つの目的であり、母数を推定するために標本から計算された値を**推定量**と言います。\n", "\n", "母集団分布について大まかなことが知りたい場合、まずは母数の中でも**母平均**$\\mu$と**母分散**$\\sigma^2$を推定することができると多くのことが分かります。\n", "\n", "例えば、ある2集団それぞれの細かい母集団分布の形が分からなくても、母平均を比較することで全体的に差があるかどうかをある程度捉えることが出来ます。\n", "\n", "あたりまえですが、母平均をいきなり求めることは出来ないので、代わりに標本サイズ$n$の標本$X_1, X_2, ... X_n$を観察し、\n", "\n", "**標本平均**$\\bar{X}=(X_1+X_2+...+X_n)/n$\n", "\n", "をまずは用います。\n", "\n", "各観測値$X_i (i=1 \\sim n)$が母集団分布に従い、その期待値が母平均$E(X_i)=\\mu$、分散が母分散$V(X_i) = \\sigma^2$とすると、\n", "\n", "標本平均の期待値$E(\\bar{X})$は\n", "\n", "$E(\\bar{X})=E((X_1+X_2+...+X_n)/n)=n\\mu/n=\\mu$\n", "\n", "となり、期待値が母平均$\\mu$と一致します。\n", "\n", "このように、標本から求めた推定量の期待値が、母数と一致するとき、その推定量は不偏性があるといい、**不偏推定量**であるといいます。\n", "\n", "さらに、標本平均の分散$V(\\bar{X})$を考えると、\n", "\n", "$V(\\bar{X}) = V(\\dfrac{1}{n}(X_1+X_2+...+X_n))=\\dfrac{1}{n^2}V(X_1+X_2+...+X_n)=n\\sigma^2/n^2=\\sigma^2/n$\n", "\n", "となり、$n\\to\\infty$のとき、標本平均の分散は0に近付き、$\\bar{X}$は$\\mu$に確率収束していきます。(**大数の法則**を意味します。)\n", "\n", "
\n", "\n", "標本分散についてですが、標本分散は、\n", "\n", "$s^2=\\dfrac{1}{n-1}\\lbrace(X_1-\\bar{X})^2+(X_2-\\bar{X})^2+...+(X_n-\\bar{X})^2\\rbrace$\n", "\n", "で定義されます。\n", "\n", "記述統計学のところでも触れた通り、$n-1$で割っているところが注意点です。\n", "\n", "これは標本分散の期待値$E(s^2)$を計算すると、\n", "\n", "$E(s^2)=\\sigma^2$と母分散$\\sigma^2$に一致するようになるからです。\n", "\n", "したがって$n-1$を使って計算された$s^2$を母分散$\\sigma^2$の不偏推定量、**不偏分散**と言います。\n", "\n", "$n-1$ではなく$n$で割った$S^2=\\dfrac{1}{n}\\lbrace(X_1-\\bar{X})^2+(X_2-\\bar{X})^2+...+(X_n-\\bar{X})^2\\rbrace$も標本分散ではありますが、**不偏でない**標本分散となります。\n", "\n", "標本平均や標本分散の様に、標本を要約し、母集団の母数のいろいろな推測に使用されるものを**統計量**と言います。\n", "\n", "今回扱った標本平均$\\bar{X}$や標本(不偏)分散$s^2$は、母平均$\\mu$や母分散$\\sigma^2$に対し、\n", "\n", "$E(\\bar{X})=\\mu$,　$E(s^2)=\\sigma^2$となるため、母集団と標本をつなぐ重要な統計量となります。\n", "\n", "
\n", "\n", "たとえば、ある実験で植物を栽培し、17個体の収穫量を測定した結果、\n", "\n", "103, 95, 123, 72, 110, 121, 136, 93, 128, 99, 100, 112, 82, 98, 106, 94, 83 (g)だったとします。\n", "\n", "この標本データから、この植物の収穫量の母平均$\\mu$、母分散$\\sigma^2$の見当をつけることを考えます。\n", "\n", "このとき標本平均と不偏分散を`mean`関数や`sd`関数で調べてみると、\n" ], "metadata": { "id": "jZS_uzq-Pa6n" } }, { "cell_type": "code", "source": [ "# 17個体の標本平均と不偏分散を求める\n", "data <- c(103, 95, 123, 72, 110, 121, 136, 93, 128, 99, 100, 112, 82, 98, 106, 94, 83)\n", "mean(data)\n", "sd(data)" ], "metadata": { "id": "MNV6g7enq08m", "colab": { "base_uri": "https://localhost:8080/", "height": 52 }, "outputId": "fe3c7223-7693-4673-a78f-ffb654b175ce" }, "execution_count": null, "outputs": [ { "output_type": "display_data", "data": { "text/html": [ "103.235294117647" ], "text/markdown": "103.235294117647", "text/latex": "103.235294117647", "text/plain": [ "[1] 103.2353" ] }, "metadata": {} }, { "output_type": "display_data", "data": { "text/html": [ "17.0899437234471" ], "text/markdown": "17.0899437234471", "text/latex": "17.0899437234471", "text/plain": [ "[1] 17.08994" ] }, "metadata": {} } ] }, { "cell_type": "markdown", "source": [ "ということが分かります。\n", "\n", "もしこの植物の母集団が正規分布している場合、正規分布の特徴から、95%程度の範囲で\n", "\n", "$\\mu\\pm2\\sigma=69\\sim137$の範囲になることが推測できます。\n", "\n", "(得られたデータもその範囲に収まっている。)\n", "\n", "この様な形で標本データから母集団の特徴を推測することができるというわけです。\n", "\n", "次回はこの部分をより細かく見ていきます。" ], "metadata": { "id": "lOAB_8-8hpkr" } }, { "cell_type": "markdown", "source": [ "## 課題\n", "\n", "今回の課題は下の課題ページを実施し、Pandaの**テスト・クイズ**に記述してください。\n", "\n", "[課題のノートブックへのリンク](https://colab.research.google.com/github/slt666666/biostatistics_text_wed/blob/main/source/_static/colab_notebook/chapter5_homework.ipynb)" ], "metadata": { "id": "688GFgWko6Lb" } } ] }