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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", mi: "mi", annotation: "annotation", mo: "mo", mfrac: "mfrac", mtext: "mtext", a: "a", msup: "msup", mn: "mn", pre: "pre", code: "code", img: "img", ul: "ul", li: "li" }, _provideComponents(), props.components); return _jsxs(_Fragment, { children: [_jsx(_components.p, { children: "今回は、中心極限定理について少し書いていこうと思います。" }), "\n", _jsx(_components.h2, { children: "中心極限定理とは" }), "\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", _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" })] }) })] }) }), "を増やせば標本平均", _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: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "μ" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mu" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" })] }) })] }) }), "に近づくことは", _jsx(_components.a, { href: "./blog1.html", children: "以前の記事" }), "で書きました(大数の法則ですね)。"] }), "\n", _jsx(_components.p, { children: "中心極限定理は、その標本平均はどのように平均に近づくのか、標本平均の分布について言及した定理になります。" }), "\n", _jsxs(_components.p, { children: ["正しく書くと、中心極限定理とは、「Xが平均 ", _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: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "μ" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mu" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" })] }) })] }) }), "(有限)、分散 ", _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: "σ" }), _jsx(_components.mn, { children: "2" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\sigma^2" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8141em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "σ" }), _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: "0.8141em" }, 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: "2" }) })] }) }) }) }) })] })] }) })] }) }), "(有限)となる確率変数は、いかなる分布でもサンプルサイズを大きくすれば、その標本平均は平均 ", _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: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "μ" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\mu" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", children: "μ" })] }) })] }) }), "、分散 ", _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.mfrac, { children: [_jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "σ" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mi, { children: "n" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\frac{\\sigma^2}{n}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.3629em", 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: "1.0179em" }, 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 mathnormal mtight", 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: "-3.394em" }, 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: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03588em" }, children: "σ" }), _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: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-2.931em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 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: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] }) })] }) }), " の正規分布に従う」です。実際に試してみましょう。"] }), "\n", _jsx(_components.h2, { children: "コイントスと中心極限定理" }), "\n", _jsx(_components.p, { children: "実際にコイン投げを3回、5回、10回投げたときの標本平均の分布を見てみましょう。" }), "\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\n", _jsx(_components.span, { className: "hljs-keyword", children: "import" }), " matplotlib.pyplot ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " plt\nnp.random.seed(", _jsx(_components.span, { className: "hljs-number", children: "1" }), ")\np3=[]\np5=[]\np10=[]\n", _jsx(_components.span, { className: "hljs-keyword", children: "for" }), " i ", _jsx(_components.span, { className: "hljs-keyword", children: "in" }), " ", _jsx(_components.span, { className: "hljs-built_in", children: "range" }), "(", _jsx(_components.span, { className: "hljs-number", children: "10000" }), "):\n p3.append(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: "3" }), ").", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "()/", _jsx(_components.span, { className: "hljs-number", children: "3" }), ")\n p5.append(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: "5" }), ").", _jsx(_components.span, { className: "hljs-built_in", children: "sum" }), "()/", _jsx(_components.span, { className: "hljs-number", children: "5" }), ")\n p10.append(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" }), ")\n\nplt.hist(p3,bins=", _jsx(_components.span, { className: "hljs-number", children: "4" }), ")\nplt.title(", _jsx(_components.span, { className: "hljs-string", children: "\"n=3\"" }), ")\nplt.show()\n\nplt.hist(p5,bins=", _jsx(_components.span, { className: "hljs-number", children: "6" }), ")\nplt.title(", _jsx(_components.span, { className: "hljs-string", children: "\"n=5\"" }), ")\nplt.show()\n\nplt.hist(p10,bins=", _jsx(_components.span, { className: "hljs-number", children: "11" }), ")\nplt.title(", _jsx(_components.span, { className: "hljs-string", children: "\"n=10\"" }), ")\nplt.show()\n"] }) }), "\n", _jsx(_components.p, { children: "結果" }), "\n", _jsx(_components.p, { children: _jsx(_components.img, { src: "/images/blog4/CLT.png", alt: "中心極限定理", title: "中心極限定理" }) }), "\n", _jsx(_components.p, { children: "nを大きくするにつれて正規分布に近くなっているのがわかります。" }), "\n", _jsx(_components.h2, { children: "大数の法則と中心極限定理の関係" }), "\n", _jsx(_components.p, { children: "nを大きくすれば、より正規分布の形に近くなることがわかりました。では、コイントスを10000回したら超きれいな正規分布の形になるに違いない！と いうことで試してみましょう。" }), "\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\n", _jsx(_components.span, { className: "hljs-keyword", children: "import" }), " matplotlib.pyplot ", _jsx(_components.span, { className: "hljs-keyword", children: "as" }), " plt\nnp.random.seed(", _jsx(_components.span, { className: "hljs-number", children: "1" }), ")\np10000=[]\n", _jsx(_components.span, { className: "hljs-keyword", children: "for" }), " i ", _jsx(_components.span, { className: "hljs-keyword", children: "in" }), " ", _jsx(_components.span, { className: "hljs-built_in", children: "range" }), "(", _jsx(_components.span, { className: "hljs-number", children: "10000" }), "):\n p10000.append(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\nplt.xlim([", _jsx(_components.span, { className: "hljs-number", children: "0" }), ",", _jsx(_components.span, { className: "hljs-number", children: "1" }), "])\nplt.hist(p10000,bins=", _jsx(_components.span, { className: "hljs-number", children: "10" }), ")\nplt.show()\n"] }) }), "\n", _jsx(_components.p, { children: "結果" }), "\n", _jsx(_components.p, { children: _jsx(_components.img, { src: "/images/blog4/CLT2.png", alt: "中心極限定理2", title: "中心極限定理2" }) }), "\n", _jsxs(_components.p, { children: ["思ってたのと違う…と思う人もいるかもしれません。しかし、中心極限定理をよくみてみると、「分散", _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.mfrac, { children: [_jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "σ" }), _jsx(_components.mn, { children: "2" })] }), _jsx(_components.mi, { children: "n" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\frac{\\sigma^2}{n}" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.3629em", 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: "1.0179em" }, 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 mathnormal mtight", 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: "-3.394em" }, 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: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03588em" }, children: "σ" }), _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: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-2.931em", marginRight: "0.0714em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.5em" } }), _jsx(_components.span, { className: "sizing reset-size3 size1 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: "0.345em" }, children: _jsx(_components.span, {}) }) })] }) }), _jsx(_components.span, { className: "mclose nulldelimiter" })] })] }) })] }) }), "」の正規分布に従うので、", _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: "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" })] }) })] }) }), "が大きくなれば分散が小さくなります。だから0.5に近いところで分布が密集しているのです。"] }), "\n", _jsxs(_components.p, { children: ["さて、勘のいい方はお気づきかもしれませんが、実はこれ、大数の弱法則と全く同じことを言っているのです(大数の弱法則については", _jsx(_components.a, { href: "./blog1.html", children: "こちら" }), "を参照)。つまり、", _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" })] }) })] }) }), "を大きくすれば、標本によるばらつきが少なくなり、標本平均は平均に近づいていくのです。"] }), "\n", _jsx(_components.h2, { children: "まとめ" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "中心極限定理とは、標本平均の分布が正規分布になる定理である。" }), 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