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arguments[0]; const {useMDXComponents: _provideComponents} = arguments[0]; function _createMdxContent(props) { const _components = Object.assign({ p: "p", h2: "h2", span: "span", math: "math", semantics: "semantics", mrow: "mrow", mo: "mo", mfrac: "mfrac", mtext: "mtext", annotation: "annotation", img: "img", mi: "mi", pre: "pre", code: "code", h3: "h3", div: "div", mn: "mn", msup: "msup", strong: "strong", msub: "msub", mtable: "mtable", mtr: "mtr", mtd: "mtd", mstyle: "mstyle", mspace: "mspace", munder: "munder", svg: "svg", path: "path", a: "a", ul: "ul", li: "li" }, _provideComponents(), props.components); return _jsxs(_Fragment, { children: [_jsx(_components.p, { children: "ブログ初投稿になります！今回は統計学の基本である大数の法則について書いていこうと思います。" }), "\n", _jsx(_components.h2, { children: "大数の法則とは" }), "\n", _jsxs(_components.p, { children: ["表と裏の出る確率が0.5であるコインがあります。このコインの試行回数を増やしたら標本平均", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mfrac, { children: [_jsx(_components.mtext, { children: "表が出る回数" }), _jsx(_components.mtext, { children: "試行回数" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "( =\\frac{表が出る回数}{試行回数} )" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.2173em", verticalAlign: "-0.345em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.8723em" }, children: [_jsxs(_components.span, { style: { top: "-2.655em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord cjk_fallback mtight", children: "試行回数" }) }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.394em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord cjk_fallback mtight", children: "表が出る回数" }) }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }) }), "は0.5に近づきそうですよね。この近づきそうっていうのを定式化したのが大数の法則と呼ばれるものです。"] }), "\n", _jsx(_components.p, { children: _jsx(_components.img, { src: "/images/blog1/blog1_2.png", alt: "コイントスと大数の法則", title: "コイントスと大数の法則" }) }), "\n", _jsxs(_components.p, { children: ["上の例を定式化してみます。確率0.5の二項分布に従う確率変数について、", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }) }), "が大きくなった時の標本平均を見てみます。"] }), "\n", _jsx(_components.pre, { children: _jsxs(_components.code, { className: "hljs language-python", children: [_jsx(_components.span, { className: "hljs-keyword", children: "import" }), " numpy ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " np\nnp.random.seed(", _jsx(_components.span, { className: "hljs-number", children: "1" }), ")\np10 = np.random.binomial(n=", _jsx(_components.span, { className: "hljs-number", children: "1" }), ",p=", _jsx(_components.span, { className: "hljs-number", children: "0.5" }), ",size=", _jsx(_components.span, { className: "hljs-number", children: "10" }), ").", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "()/", _jsx(_components.span, { className: "hljs-number", children: "10" }), "\np100 = np.random.binomial(n=", _jsx(_components.span, { className: "hljs-number", children: "1" }), ",p=", _jsx(_components.span, { className: "hljs-number", children: "0.5" }), ",size=", _jsx(_components.span, { className: "hljs-number", children: "100" }), ").", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "()/", _jsx(_components.span, { className: "hljs-number", children: "100" }), "\np1000 = np.random.binomial(n=", _jsx(_components.span, { className: "hljs-number", children: "1" }), ",p=", _jsx(_components.span, { className: "hljs-number", children: "0.5" }), ",size=", _jsx(_components.span, { className: "hljs-number", children: "1000" }), ").", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "()/", _jsx(_components.span, { className: "hljs-number", children: "1000" }), "\np10000 = np.random.binomial(n=", _jsx(_components.span, { className: "hljs-number", children: "1" }), ",p=", _jsx(_components.span, { className: "hljs-number", children: "0.5" }), ",size=", _jsx(_components.span, { className: "hljs-number", children: "10000" }), ").", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "()/", _jsx(_components.span, { className: "hljs-number", children: "10000" }), "\n", _jsx(_components.span, { className: "hljs-built_in", children: "print" }), "(", _jsx(_components.span, { className: "hljs-string", children: "\"n=10 : {}\\nn=100 : {}\\nn=1000 : {}\\nn=10000 : {}\"" }), ".", _jsx(_components.span, { className: "hljs-built_in", children: "format" }), "(p10,p100,p1000,p10000))\n"] }) }), "\n", _jsx(_components.p, { children: "結果" }), "\n", _jsx(_components.pre, { children: _jsx(_components.code, { className: "hljs language-text", children: "n=10 : 0.2\nn=100 : 0.54\nn=1000 : 0.511\nn=10000 : 0.5019\n" }) }), "\n", _jsx(_components.p, { children: "nを大きくすれば確かに0.5に近くなります。当たり前っちゃ当たり前。" }), "\n", _jsx(_components.h2, { children: "色々な分布での大数の法則" }), "\n", _jsx(_components.p, { children: "上記は二項分布に従う確率変数についての大数の法則を見ましたが、さまざまな分布について大数の法則を見ていきましょう！" }), "\n", _jsx(_components.h3, { children: "ポアソン分布" }), "\n", _jsx(_components.p, { children: "1日に平均5個不良品が存在する部品があったとすると、1日に不良品が存在する個数の確率はパラメータ5のポアソン分布に従います。これを何日か続けていった場合、その個数の平均は5に近づくでしょうか？試してみましょう。" }), "\n", _jsx(_components.pre, { children: _jsxs(_components.code, { className: "hljs language-python", children: [_jsx(_components.span, { className: "hljs-keyword", children: "import" }), " numpy ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " np\nnp.random.seed(", _jsx(_components.span, { className: "hljs-number", children: "1" }), ")\np10 = np.random.poisson(", _jsx(_components.span, { className: "hljs-number", children: "5" }), ",", _jsx(_components.span, { className: "hljs-number", children: "10" }), ").", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "()/", _jsx(_components.span, { className: "hljs-number", children: "10" }), "\np100 = np.random.poisson(", _jsx(_components.span, { className: "hljs-number", children: "5" }), ",", _jsx(_components.span, { className: "hljs-number", children: "100" }), ").", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "()/", _jsx(_components.span, { className: "hljs-number", children: "100" }), "\np1000 = np.random.poisson(", _jsx(_components.span, { className: "hljs-number", children: "5" }), ",", _jsx(_components.span, { className: "hljs-number", children: "1000" }), ").", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "()/", _jsx(_components.span, { className: "hljs-number", children: "1000" }), "\np10000 = np.random.poisson(", _jsx(_components.span, { className: "hljs-number", children: "5" }), ",", _jsx(_components.span, { className: "hljs-number", children: "10000" }), ").", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "()/", _jsx(_components.span, { className: "hljs-number", children: "10000" }), "\n", _jsx(_components.span, { className: "hljs-built_in", children: "print" }), "(", _jsx(_components.span, { className: "hljs-string", children: "\"n=10 : {}\\nn=100 : {}\\nn=1000 : {}\\nn=10000 : {}\"" }), ".", _jsx(_components.span, { className: "hljs-built_in", children: "format" }), "(p10,p100,p1000,p10000))\n"] }) }), "\n", _jsx(_components.p, { children: "結果" }), "\n", _jsx(_components.pre, { children: _jsx(_components.code, { className: "hljs language-text", children: "n=10 : 4.1\nn=100 : 5.15\nn=1000 : 4.994\nn=10000 : 4.9779\n" }) }), "\n", _jsx(_components.p, { children: "ちゃんと5に近づいてますね。" }), "\n", _jsx(_components.h3, { children: "正規分布" }), "\n", _jsx(_components.p, { children: "みんな大好き正規分布ですね。平均0、分散1の正規分布から生成された確率変数を何個か生成したら、その標本平均は0に近づくでしょうか？" }), "\n", _jsx(_components.pre, { children: _jsxs(_components.code, { className: "hljs language-python", children: [_jsx(_components.span, { className: "hljs-keyword", children: "import" }), " numpy ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " np\nnp.random.seed(", _jsx(_components.span, { className: "hljs-number", children: "1" }), ")\np10 = np.random.normal(", _jsx(_components.span, { className: "hljs-number", children: "0" }), ",", _jsx(_components.span, { className: "hljs-number", children: "1" }), ",", _jsx(_components.span, { className: "hljs-number", children: "10" }), ").", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "()/", _jsx(_components.span, { className: "hljs-number", children: "10" }), "\np100 = np.random.normal(", _jsx(_components.span, { className: "hljs-number", children: "0" }), ",", _jsx(_components.span, { className: "hljs-number", children: "1" }), ",", _jsx(_components.span, { className: "hljs-number", children: "100" }), ").", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "()/", _jsx(_components.span, { className: "hljs-number", children: "100" }), "\np1000 = np.random.normal(", _jsx(_components.span, { className: "hljs-number", children: "0" }), ",", _jsx(_components.span, { className: "hljs-number", children: "1" }), ",", _jsx(_components.span, { className: "hljs-number", children: "1000" }), ").", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "()/", _jsx(_components.span, { className: "hljs-number", children: "1000" }), "\np10000 = np.random.normal(", _jsx(_components.span, { className: "hljs-number", children: "0" }), ",", _jsx(_components.span, { className: "hljs-number", children: "1" }), ",", _jsx(_components.span, { className: "hljs-number", children: "10000" }), ").", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "()/", _jsx(_components.span, { className: "hljs-number", children: "10000" }), "\n", _jsx(_components.span, { className: "hljs-built_in", children: "print" }), "(", _jsx(_components.span, { className: "hljs-string", children: "\"n=10 : {}\\nn=100 : {}\\nn=1000 : {}\\nn=10000 : {}\"" }), ".", _jsx(_components.span, { className: "hljs-built_in", children: "format" }), "(p10,p100,p1000,p10000))\n"] }) }), "\n", _jsx(_components.p, { children: "結果" }), "\n", _jsx(_components.pre, { children: _jsx(_components.code, { className: "hljs language-text", children: "n=10 : -0.09714089080609985\nn=100 : 0.07431865897973937\nn=1000 : 0.03360117004534288\nn=10000 : 0.009015266856491327\n" }) }), "\n", _jsx(_components.p, { children: "標本平均が0に近づき、大数の法則が成り立つことがわかります。" }), "\n", _jsx(_components.h2, { children: "大数の法則が成り立たないケース" }), "\n", _jsx(_components.p, { children: "さて、今までの例では標本平均が母平均に近づくことを確認しました。では、どんな確率分布でも大数の法則は成り立つのでしょうか？" }), "\n", _jsx(_components.p, { children: "実は、成り立たない例があります。具体的な例として、コーシー分布があります。コーシー分布とは、標準正規分布同士をわり算した分布です。確率密度関数の形は正規分布に似ていますが、裾が広いのが特徴です。" }), "\n", _jsx(_components.p, { children: "コーシー分布の確率密度関数 :" }), "\n", _jsx(_components.div, { className: "math math-display", children: _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "f" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "x" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mfrac, { children: [_jsx(_components.mn, { children: "1" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "π" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { children: "+" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "x" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "f(x)=\\frac{1}{\\pi(1+x^2)}" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.10764em" }, children: "f" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.2574em", verticalAlign: "-0.936em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "π" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "x" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7401em" }, children: _jsxs(_components.span, { style: { top: "-2.989em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.936em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] })] }) }) }), "\n", _jsx(_components.p, { children: "グラフを書くと以下のようになります。正規分布と比べるとかなり裾が広いです。" }), "\n", _jsx(_components.pre, { children: _jsxs(_components.code, { className: "hljs language-python", children: [_jsx(_components.span, { className: "hljs-keyword", children: "import" }), " matplotlib.pyplot ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " plt\n", _jsx(_components.span, { className: "hljs-comment", children: "#関数定義" }), "\n", _jsx(_components.span, { className: "hljs-keyword", children: "def" }), " ", _jsx(_components.span, { className: "hljs-title function_", children: "normal" }), "(", _jsx(_components.span, { className: "hljs-params", children: "x" }), "):\n ", _jsx(_components.span, { className: "hljs-keyword", children: "return" }), " (np.exp(-x**", _jsx(_components.span, { className: "hljs-number", children: "2" }), "/", _jsx(_components.span, { className: "hljs-number", children: "2" }), ")) /np.sqrt(", _jsx(_components.span, { className: "hljs-number", children: "2" }), "*np.pi)\n", _jsx(_components.span, { className: "hljs-keyword", children: "def" }), " ", _jsx(_components.span, { className: "hljs-title function_", children: "cauchy" }), "(", _jsx(_components.span, { className: "hljs-params", children: "x" }), "):\n ", _jsx(_components.span, { className: "hljs-keyword", children: "return" }), " ", _jsx(_components.span, { className: "hljs-number", children: "1" }), "/(np.pi*(", _jsx(_components.span, { className: "hljs-number", children: "1" }), "+x**", _jsx(_components.span, { className: "hljs-number", children: "2" }), "))\n", _jsx(_components.span, { className: "hljs-comment", children: "#分布作成" }), "\nx = np.arange(-", _jsx(_components.span, { className: "hljs-number", children: "5" }), ",", _jsx(_components.span, { className: "hljs-number", children: "5" }), ",", _jsx(_components.span, { className: "hljs-number", children: "0.1" }), ")\ny_normal = np.vectorize(normal)(x)\ny_cauchy = np.vectorize(cauchy)(x)\n", _jsx(_components.span, { className: "hljs-comment", children: "#描画" }), "\nplt.plot(x,y_normal,label=", _jsx(_components.span, { className: "hljs-string", children: "\"normal\"" }), ")\nplt.plot(x,y_cauchy,label=", _jsx(_components.span, { className: "hljs-string", children: "\"cauchy\"" }), ")\nplt.legend()\n"] }) }), "\n", _jsx(_components.p, { children: "結果" }), "\n", _jsx(_components.p, { children: _jsx(_components.img, { src: "/images/blog1/blog1_1.png", alt: "正規分布とコーシー分布", title: "正規分布とコーシー分布" }) }), "\n", _jsx(_components.p, { children: "では実際にコーシー分布に従う確率変数の標本平均が0に近づくか見てみましょう。" }), "\n", _jsx(_components.pre, { children: _jsxs(_components.code, { className: "hljs language-python", children: [_jsx(_components.span, { className: "hljs-keyword", children: "import" }), " numpy ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " np\nnp.random.seed(", _jsx(_components.span, { className: "hljs-number", children: "1" }), ")\np10 = np.random.standard_cauchy(size=", _jsx(_components.span, { className: "hljs-number", children: "10" }), ").", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "()/", _jsx(_components.span, { className: "hljs-number", children: "10" }), "\np100 = np.random.standard_cauchy(size=", _jsx(_components.span, { className: "hljs-number", children: "100" }), ").", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "()/", _jsx(_components.span, { className: "hljs-number", children: "100" }), "\np1000 = np.random.standard_cauchy(size=", _jsx(_components.span, { className: "hljs-number", children: "1000" }), ").", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "()/", _jsx(_components.span, { className: "hljs-number", children: "1000" }), "\np10000 = np.random.standard_cauchy(size=", _jsx(_components.span, { className: "hljs-number", children: "10000" }), ").", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "()/", _jsx(_components.span, { className: "hljs-number", children: "10000" }), "\n", _jsx(_components.span, { className: "hljs-built_in", children: "print" }), "(", _jsx(_components.span, { className: "hljs-string", children: "\"n=10 : {}\\nn=100 : {}\\nn=1000 : {}\\nn=10000 : {}\"" }), ".", _jsx(_components.span, { className: "hljs-built_in", children: "format" }), "(p10,p100,p1000,p10000))\n"] }) }), "\n", _jsx(_components.p, { children: "結果" }), "\n", _jsx(_components.pre, { children: _jsx(_components.code, { className: "hljs language-text", children: "n=10 : -0.6742674056116752\nn=100 : 0.4863366954564788\nn=1000 : -6.369291105541214\nn=10000 : 2.640367957638925\n" }) }), "\n", _jsxs(_components.p, { children: ["全然0に近づいていませんね。これはコーシー分布は裾が長く外れ値を取りやすいために起こります。一般に、大数の法則は", _jsx(_components.strong, { children: "平均が有限な確率分布に従う独立同分布な確率変数" }), "に対して適用できます。(コーシー分布は実は平均を持ちません。)"] }), "\n", _jsx(_components.h2, { children: "収束の種類" }), "\n", _jsx(_components.p, { children: "ここから難易度が急激に上がります。今まで「近づく」と曖昧に書いていましたが、ここからは数学的に「近づく」を考えていきます。" }), "\n", _jsx(_components.p, { children: "まず、コイン投げにおける確率変数を考えましょう。結論からかくと以下の様になります。" }), "\n", _jsx(_components.div, { className: "math math-display", children: _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "ω" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "{" }), _jsxs(_components.mtable, { rowspacing: "0.36em", columnalign: "left left", columnspacing: "1em", children: [_jsx(_components.mtr, { children: _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "1" }), _jsx(_components.mspace, { width: "2.8453em" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mtext, { children: " 回目で表が出る" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }) }) }) }), _jsx(_components.mtr, { children: _jsx(_components.mtd, { children: _jsx(_components.mstyle, { scriptlevel: "0", displaystyle: "false", children: _jsxs(_components.mrow, { children: [_jsx(_components.mn, { children: "0" }), _jsx(_components.mspace, { width: "2.8453em" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "i" }), _jsx(_components.mtext, { children: " 回目で裏が出る" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }) }) }) })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: " X_i(\\omega)= \\begin{cases} 1 \\hspace{1cm} (i \\text{ 回目で表が出る}) \\\\ 0 \\hspace{1cm} (i \\text{ 回目で裏が出る}) \\end{cases} " })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0785em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "3em", verticalAlign: "-1.25em" } }), _jsxs(_components.span, { className: "minner", children: [_jsx(_components.span, { className: "mopen delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size4", children: "{" }) }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mtable", children: _jsx(_components.span, { className: "col-align-l", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.69em" }, children: [_jsxs(_components.span, { style: { top: "-3.69em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.008em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "2.8453em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsxs(_components.span, { className: "mord text", children: [_jsx(_components.span, { className: "mord", children: " " }), _jsx(_components.span, { className: "mord cjk_fallback", children: "回目で表が出る" })] }), _jsx(_components.span, { className: "mclose", children: ")" })] })] }), _jsxs(_components.span, { style: { top: "-2.25em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3.008em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "0" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "2.8453em" } }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", children: "i" }), _jsxs(_components.span, { className: "mord text", children: [_jsx(_components.span, { className: "mord", children: " " }), _jsx(_components.span, { className: "mord cjk_fallback", children: "回目で裏が出る" })] }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "1.19em" }, children: _jsx(_components.span, {}) }) })] }) }) }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] })] }) }) }), "\n", _jsxs(_components.p, { children: ["上記の ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "ω" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\omega" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" })] }) })] }) }), " は", _jsx(_components.strong, { children: "標本点" }), "と呼ばれ、今回は（表、表、裏、…）等の表裏の組み合わせとして定義します。"] }), "\n", _jsxs(_components.p, { children: ["次に、大数の法則を数式で書いてみます。", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "{" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { children: "i" })] }), _jsxs(_components.msub, { children: [_jsx(_components.mo, { stretchy: "false", children: "}" }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "i" }), _jsx(_components.mo, { children: "∈" }), _jsx(_components.mi, { mathvariant: "double-struck", children: "N" })] })] })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\{X_i\\}_{i\\in \\mathbb{N}}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mopen", children: "{" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0785em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsxs(_components.span, { className: "mclose", children: [_jsx(_components.span, { className: "mclose", children: "}" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3322em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }), _jsx(_components.span, { className: "mrel mtight", children: "∈" }), _jsx(_components.span, { className: "mord mathbb mtight", children: "N" })] }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.1774em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }) }), " が独立同分布で ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "X" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "X" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" })] }) })] }) }), " の分布に従うとします。", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "X" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "X" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" })] }) })] }) }), " の平均が有限のとき、"] }), "\n", _jsx(_components.div, { className: "math math-display", children: _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.munder, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "lim" }), _jsx(_components.mo, { children: "⁡" })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] })] }), _jsxs(_components.mfrac, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "ω" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "+" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "ω" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mo, { children: "⋯" }), _jsx(_components.mo, { children: "+" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "ω" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "E" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "X" }), _jsx(_components.mo, { stretchy: "false", children: "]" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: " \\lim_{n\\rightarrow \\infty}\\frac{X_1(\\omega)+X_2(\\omega)+\\cdots+X_n(\\omega)}{n}=E[X]" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.127em", verticalAlign: "-0.7em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.6944em" }, children: [_jsxs(_components.span, { style: { top: "-2.4em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "→" }), _jsx(_components.span, { className: "mord mtight", children: "∞" })] }) })] }), _jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop", children: "lim" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.427em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", children: "n" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0785em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0785em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "minner", children: "⋯" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0785em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "E" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "mclose", children: "]" })] })] })] }) }) }), "\n", _jsx(_components.p, { children: "と書くことができます。" }), "\n", _jsxs(_components.p, { children: ["さて、この式は全ての ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "ω" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\omega" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" })] }) })] }) }), " について満たしているでしょうか？答えは否です。なぜかと言いますと、もし ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "ω" }), _jsx(_components.mo, { mathvariant: "normal", lspace: "0em", rspace: "0em", children: "′" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\omega'" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.7519em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7519em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "′" }) }) })] }) }) }) }) })] })] }) })] }) }), " が全て「表」とした場合、全ての ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "i" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "i" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6595em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "i" })] }) })] }) }), " について ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { children: "i" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "ω" }), _jsx(_components.mo, { mathvariant: "normal", lspace: "0em", rspace: "0em", children: "′" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "X_i(\\omega')=1" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.0019em", verticalAlign: "-0.25em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3117em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0785em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "i" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7519em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "′" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }) }), " となるので、以下のようになります。"] }), "\n", _jsx(_components.div, { className: "math math-display", children: _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.munder, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "lim" }), _jsx(_components.mo, { children: "⁡" })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] })] }), _jsxs(_components.mfrac, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "ω" }), _jsx(_components.mo, { mathvariant: "normal", lspace: "0em", rspace: "0em", children: "′" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "+" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "ω" }), _jsx(_components.mo, { mathvariant: "normal", lspace: "0em", rspace: "0em", children: "′" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mo, { children: "⋯" }), _jsx(_components.mo, { children: "+" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "ω" }), _jsx(_components.mo, { mathvariant: "normal", lspace: "0em", rspace: "0em", children: "′" })] }), _jsx(_components.mo, { 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style: { top: "-2.55em", marginLeft: "-0.0785em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7519em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "′" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "minner", children: "⋯" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0785em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7519em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: _jsx(_components.span, { className: "mord mtight", children: "′" }) }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsxs(_components.span, { className: "mrel", children: [_jsx(_components.span, { className: "mrel", children: _jsx(_components.span, { className: "mord vbox", children: _jsx(_components.span, { className: "thinbox", children: _jsxs(_components.span, { className: "rlap", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8889em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "inner", children: _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mrel", children: "" }) }) }), _jsx(_components.span, { className: "fix" })] }) }) }) }), _jsx(_components.span, { className: "mrel", children: "=" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.0074em", verticalAlign: "-0.686em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.3214em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "2" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord", children: "1" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] })] })] }) }) }), "\n", _jsxs(_components.p, { children: ["つまり、", _jsxs(_components.strong, { children: [_jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "ω" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\omega" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" })] }) })] }) }), "がどの程度(1)を満たしているか"] }), "が重要になってきます。実は、どの程度満たすかによって、大数の法則の呼ばれ方が異なってきます。大数の法則には2種類あります！"] }), "\n", _jsx(_components.h3, { children: "大数の強法則" }), "\n", _jsxs(_components.p, { children: ["まずは強法則から。端的に言ってしまうと、(1)を満たさない ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "ω" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\omega" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" })] }) })] }) }), " の集合の確率は0ですよ、というのが強法則です。確率0ってなんだと思われますが、無限回コイン投げを実施して全て表の確率が0であることは計算すればわかります。これを(1)を満たさない ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "ω" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\omega" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" })] }) })] }) }), " の集合全体で考えても確率が0になります。式で書くと以下のようになります。"] }), "\n", _jsx(_components.div, { className: "math math-display", children: _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "P" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { fence: "true", children: "(" }), _jsxs(_components.munder, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "lim" }), _jsx(_components.mo, { children: "⁡" })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] })] }), _jsxs(_components.mfrac, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "ω" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "+" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "ω" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { 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\\infty}\\frac{X_1(\\omega)+X_2(\\omega)+\\cdots+X_n(\\omega)}{n}=E[X]\\right)=1 " })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.4em", verticalAlign: "-0.95em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "P" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "minner", children: [_jsx(_components.span, { className: "mopen delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size3", children: "(" }) }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.6944em" }, children: [_jsxs(_components.span, { style: { top: "-2.4em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "→" }), _jsx(_components.span, { className: "mord mtight", children: "∞" })] }) })] }), _jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop", children: "lim" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.427em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", children: "n" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0785em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0785em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "minner", children: "⋯" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0785em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "E" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mclose delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size3", children: ")" }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "1" })] })] })] }) }) }), "\n", _jsx(_components.p, { children: "また、次のようにも書きます。" }), "\n", _jsx(_components.div, { className: "math math-display", children: _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.munder, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "lim" }), _jsx(_components.mo, { children: "⁡" })] }), _jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "n" }), _jsx(_components.mo, { children: "→" }), _jsx(_components.mi, { mathvariant: "normal", children: "∞" })] })] }), _jsxs(_components.mfrac, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "ω" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "+" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "ω" }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "+" }), _jsx(_components.mo, { children: "⋯" }), _jsx(_components.mo, { children: "+" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "X" }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "ω" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.mi, { children: "n" })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mi, { children: "E" }), _jsx(_components.mo, { stretchy: "false", children: "[" }), _jsx(_components.mi, { children: "X" }), _jsx(_components.mo, { stretchy: "false", children: "]" }), _jsx(_components.mspace, { width: "2em" }), _jsx(_components.mi, { children: "a" }), _jsx(_components.mi, { mathvariant: "normal", children: "." }), _jsx(_components.mi, { children: "s" }), _jsx(_components.mi, { mathvariant: "normal", children: "." })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: " \\lim_{n\\rightarrow \\infty}\\frac{X_1(\\omega)+X_2(\\omega)+\\cdots+X_n(\\omega)}{n}=E[X] \\qquad a.s. " })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "2.127em", verticalAlign: "-0.7em" } }), _jsx(_components.span, { className: "mop op-limits", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "0.6944em" }, children: [_jsxs(_components.span, { style: { top: "-2.4em", marginLeft: "0em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }), _jsx(_components.span, { className: "mrel mtight", children: "→" }), _jsx(_components.span, { className: "mord mtight", children: "∞" })] }) })] }), _jsxs(_components.span, { style: { top: "-3em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { children: _jsx(_components.span, { className: "mop", children: "lim" }) })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.7em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.1667em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mopen nulldelimiter" }), _jsx(_components.span, { className: "mfrac", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsxs(_components.span, { className: "vlist", style: { height: "1.427em" }, children: [_jsxs(_components.span, { style: { top: "-2.314em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "mord", children: _jsx(_components.span, { className: "mord mathnormal", children: "n" }) })] }), _jsxs(_components.span, { style: { top: "-3.23em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsx(_components.span, { className: "frac-line", style: { borderBottomWidth: "0.04em" } })] }), _jsxs(_components.span, { style: { top: "-3.677em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "3em" } }), _jsxs(_components.span, { className: "mord", children: [_jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0785em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0785em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "minner", children: "⋯" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "+" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.1514em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0785em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mathnormal mtight", children: "n" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.686em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "E" }), _jsx(_components.span, { className: "mopen", children: "[" }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07847em" }, children: "X" }), _jsx(_components.span, { className: "mclose", children: "]" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "2em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "a" }), _jsx(_components.span, { className: "mord", children: "." }), _jsx(_components.span, { className: "mord mathnormal", children: "s" }), _jsx(_components.span, { className: "mord", children: "." })] })] })] }) }) }), "\n", _jsxs(_components.p, { children: ["「a.s.」はalmost surelyの略で、「ほとんど確実に」という意味になります。この収束の種類を", _jsx(_components.strong, { children: "概収束" }), "と言ったりします。"] }), "\n", _jsx(_components.h3, { children: "大数の弱法則" }), "\n", _jsx(_components.p, { children: "実は大数の強法則を証明するのって結構難しいんです。ということで、もう少し条件を緩くしたものが大数の弱法則です。定義は以下のようになります。" }), "\n", _jsx(_components.div, { className: "math math-display", children: _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "∀" }), _jsx(_components.mi, { children: "ε" }), _jsx(_components.mo, { children: ">" }), _jsx(_components.mn, { children: "0" }), _jsx(_components.mo, { separator: "true", children: "," }), _jsx(_components.mspace, { width: "2.8453em" }), _jsxs(_components.munder, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { 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v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v1200 v585 h43z" }) }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "​" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.95em" }, children: _jsx(_components.span, {}) }) })] }) }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "≥" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "ε" }), _jsx(_components.span, { className: "mclose delimcenter", style: { top: "0em" }, children: _jsx(_components.span, { className: "delimsizing size3", children: ")" }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "0" })] })] })] }) }) }), "\n", _jsxs(_components.p, { children: ["この式は、", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }) }), " を大きくしていくと標本平均は標本によるばらつきが小さくなり、値が期待値に近くなることを表しています。大数の強法則とは ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "n" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "n" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "n" })] }) })] }) }), " を無限にする位置が異なっていますね。この収束を", _jsx(_components.strong, { children: "確率収束" }), "と言います。"] }), "\n", _jsxs(_components.p, { children: ["大数の弱法則は中心極限定理と関係があります。もし詳しく知りたければ、", _jsx(_components.a, { href: "./blog4", children: "こちら" }), "も参考にしてください。"] }), "\n", _jsx(_components.h2, { children: "まとめ" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "大数の法則は標本平均が期待値に収束する定理である。" }), "\n", _jsx(_components.li, { children: "大数の法則は平均を持たない確率変数に対して成り立つとは限らない。" }), "\n", _jsxs(_components.li, { children: ["大数の法則には2種類あり、どの程度の ", _jsx(_components.span, { className: "math math-inline", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "ω" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\omega" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "ω" })] }) })] }) }), " で(1)が成り立つかで変わってくる。"] }), "\n"] })] }); } function MDXContent(props = {}) { const {wrapper: MDXLayout} = Object.assign({}, _provideComponents(), props.components); return MDXLayout ? _jsx(MDXLayout, Object.assign({}, props, { children: _jsx(_createMdxContent, props) })) : _createMdxContent(props); } return { default: MDXContent }; a:["$","div",null,{"className":"container mx-auto px-4 py-8 max-w-7xl","children":["$","div",null,{"className":"flex flex-col md:flex-row gap-8","children":[["$","article",null,{"className":"w-full md:w-[70%] bg-white shadow-lg rounded-lg 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