\u5e7e\u4f55\u5206\u5e03\u306e\u6982\u8981<\/strong><\/p>\r\n\u5e7e\u4f55\u5206\u5e03\u3068\u306f\u300c\u6210\u529f\u78ba\u7387\u304c\\(p\\)\u3067\u3042\u308b\u72ec\u7acb\u306a\u30d9\u30eb\u30cc\u30fc\u30a4\u8a66\u884c\u3092\u7e70\u308a\u8fd4\u3059\u3068\u304d\u3001\u521d\u3081\u3066\u6210\u529f\u3059\u308b\u307e\u3067\u306e\u8a66\u884c\u56de\u6570\\(X \\)\u304c\u5f93\u3046\u78ba\u7387\u5206\u5e03\u300d\u306e\u3053\u3068\u3067\u3059\u3002<\/p>\r\n
\u5e7e\u4f55\u5206\u5e03\u306b\u5f93\u3046\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u78ba\u7387\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\r\n
\\[P(X=k)=(1-p)^{k-1}p \\qquad (k=1,2,3, \\ldots)\\]<\/p>\r\n
\u306f\u3058\u3081\u306e\\((k-1)\\)\u56de\u306e\u8a66\u884c\u3067\u5931\u6557\u3057\u7d9a\u3051\u3001\\(k\\)\u56de\u76ee\u306e\u8a66\u884c\u3067\u521d\u3081\u3066\u6210\u529f\u3059\u308b\u78ba\u7387\u3092\u8868\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\r\n
\u3088\u308a\u8a73\u7d30\u306b\u7406\u89e3\u3057\u305f\u3044\u5834\u5408\u306f\u3001\u4f8b\u3048\u3070\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u304c\u53c2\u8003\u306b\u306a\u308a\u307e\u3059\u3002[blogcard url=”https:\/\/bellcurve.jp\/statistics\/course\/6988.html?srsltid=AfmBOopRKi1XQ8UQZNgZEIiXmtanurH2K8o-r9S7VoFkrw7BNyfk-ztV”]<\/p>\r\n
<\/p>\n\n
\u30ab\u30b9\u30df\u306e\u78ba\u7387\u30e2\u30c7\u30eb<\/h2>\n
\u5e7e\u4f55\u5206\u5e03\u306b\u3064\u3044\u3066\u5927\u307e\u304b\u306b\u7406\u89e3\u3057\u305f\u3068\u3053\u308d\u3067\u3001\u30ab\u30b9\u30df\u304c\u30b3\u30a4\u30f3\u30c8\u30b9\u3059\u308b\u8a66\u884c\u3092\u5b9a\u5f0f\u5316\u3057\u307e\u3059\u3002<\/p>\r\n
\u8a2d\u5b9a<\/span>
\r\n<\/strong>\u3000\u30fb\u3000\u30b3\u30a4\u30f3\u306e\u8868\u304c\u51fa\u308b\u78ba\u7387\u3092\\(p\\)\u3068\u3059\u308b
\r\n\u3000\u30fb\u30001\u679a\u306e\u30ab\u30b9\u30df\u304c\u30b3\u30a4\u30f3\u30c8\u30b9\u3092\u884c\u3046\u56de\u6570\uff08\u521d\u3081\u3066\u88cf\u304c\u51fa\u308b\u8a66\u884c\u3082\u542b\u3080\uff09\u3092\\(X\\)\u3068\u3059\u308b<\/p>\r\n <\/p>\r\n
\u91cd\u8981\u306a\u4eee\u5b9a<\/span><\/span>
\r\n<\/strong>\u30ab\u30b9\u30df\u306b\u3088\u308b\u5404\u30b3\u30a4\u30f3\u30c8\u30b9\u306f\u72ec\u7acb\u306b\u540c\u4e00\u306a\u30d9\u30eb\u30cc\u30fc\u30a4\u5206\u5e03\u306b\u5f93\u3046<\/strong>\u3053\u3068\u3092\u4eee\u5b9a\u3057\u307e\u3059\u3002\u7c21\u5358\u306b\u8a00\u3048\u3070\u3001\u3044\u304b\u306a\u308b\u5834\u5408\u3067\u3082\u30b3\u30a4\u30f3\u306e\u8868\u304c\u51fa\u308b\u78ba\u7387\u306f\u7b49\u3057\u304f\\(p\\)\u3067\u3042\u308b<\/strong>\u3068\u3044\u3046\u4eee\u5b9a\u3067\u3059\u3002<\/p>\r\n\u3064\u307e\u308a\u3001\u6b21\u306e\u3088\u3046\u306a\u53ef\u80fd\u6027\u306f\u7121\u8996\u3059\u308b\u3082\u306e\u3068\u3057\u307e\u3059\u3002
\r\n\u3000\u30fb\u30d7\u30ec\u30a4\u30e4\u30fc\u304c\u610f\u56f3\u7684\u306b\u8868\u3092\u51fa\u3059\u3053\u3068\u304c\u3067\u304d\u308b
\r\n\u3000\u30fb\u8868\u304c\u51fa\u7d9a\u3051\u308b\u307b\u3069\u3001\u8868\u304c\u51fa\u308b\u78ba\u7387\u304c\u4f4e\u304f\u306a\u3063\u3066\u3044\u304f
\r\n\u3000\u30fb\u30b3\u30a4\u30f3\u30c8\u30b9\u306e\u56de\u6570\u306b\u4e0a\u9650\u304c\u8a2d\u5b9a\u3055\u308c\u3066\u3044\u308b
\r\n\u3000\u30fb\u30ab\u30b9\u30df\u3092\u4f7f\u3046\u30bf\u30fc\u30f3\u306b\u3088\u3063\u3066\u8868\u306e\u78ba\u7387\u304c\u5909\u308f\u308b<\/p>\r\n
<\/p>\r\n
\u4e0a\u8a18\u306e\u4eee\u5b9a\u306e\u4e0b\u3067\u306f\u3001\u6b21\u306e\u3053\u3068\u304c\u3044\u3048\u307e\u3059\u3002
\r\n\u3000\u30fb\\(X\\)\u306f\u5e7e\u4f55\u5206\u5e03\u306b\u5f93\u3046
\r\n\u3000\u30fb(1\u679a\u306e\u30ab\u30b9\u30df\u304c\u7372\u5f97\u3059\u308b\u6c34\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u500b\u6570) \\(=X-1\\)<\/p>\r\n
<\/p>\r\n
\u78ba\u7387\u95a2\u6570
\r\n<\/strong><\/span>1\u679a\u306e\u30ab\u30b9\u30df\u304c\u30b3\u30a4\u30f3\u30c8\u30b9\u3092\u884c\u3046\u56de\u6570\\(X\\)\u306e\u78ba\u7387\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u307e\u3059\uff1a<\/p>\r\n$$ P(X=k)=p^{k-1}(1-p) \\qquad (k=1,2,3, \\ldots) $$<\/p>\r\n
\u306f\u3058\u3081\u306e\\((k-1)\\)\u56de\u306e\u30b3\u30a4\u30f3\u30c8\u30b9\u3067\u8868\u3092\u51fa\u3057\u7d9a\u3051\u3001\\(k\\)\u56de\u76ee\u3067\u521d\u3081\u3066\u88cf\u3092\u51fa\u3059\u78ba\u7387\u3092\u8868\u3057\u3066\u3044\u307e\u3059\u3002
\r\n\u203b\u4e00\u822c\u7684\u306a\u5e7e\u4f55\u5206\u5e03\u306e\u78ba\u7387\u95a2\u6570\u3068\u306f\\(p\\), \\(1-p\\)\u304c\u9006\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\r\n
<\/p>\n\n
\u30c7\u30fc\u30bf\u53ce\u96c6<\/h2>\n
\u96c6\u8a08\u65b9\u6cd5<\/span><\/span>
\r\n<\/strong>\u3000\u30fb\u30b5\u30f3\u30d7\u30eb\u30b5\u30a4\u30ba\uff1a\\(N=100\\)
\r\n\u3000\u30fb\u96c6\u8a08\u671f\u9593\uff1a2024\/11\/21~2024\/11\/24
\r\n\u3000\u30fb\u96c6\u8a08\u5834\u6240\uff1a\u30aa\u30f3\u30e9\u30a4\u30f3\u5bfe\u6226\uff08\u3059\u3079\u3066\u300c\u6700\u5f37\u306e\u907a\u4f1d\u5b50 \u30a8\u30f3\u30d6\u30ec\u30e0\u30a4\u30d9\u30f3\u30c81\u300d\u3067\u96c6\u8a08
\r\n![\"\"](\"https:\/\/since2020.jp\/media\/wp-content\/uploads\/2024\/11\/poke_event.png\")
<\/p>\r\n <\/p>\r\n
\u30c7\u30fc\u30bf\u306e\u6982\u8981<\/span><\/span>
\r\n<\/strong>\u30aa\u30f3\u30e9\u30a4\u30f3\u5bfe\u6226\u3067\u53ce\u96c6\u3057\u305f100\u679a\u306e\u30ab\u30b9\u30df\u306b\u3088\u308b\u30b3\u30a4\u30f3\u30c8\u30b9\u306e\u7d50\u679c\u3092\u7c21\u5358\u306b\u7d39\u4ecb\u3057\u307e\u3059\u3002<\/p>\r\n\u5b9f\u969b\u306b\u96c6\u8a08\u3055\u308c\u305f\u300c1\u679a\u306e\u30ab\u30b9\u30df\u304c\u30b3\u30a4\u30f3\u30c8\u30b9\u3092\u884c\u3063\u305f\u56de\u6570\u300d\u3092 \\(x_i\\ (i=1,2,…,N)\\) \u3068\u8868\u3059\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\r\n
\u30ab\u30b9\u30df\u304c\u7372\u5f97\u3057\u305f\u6c34\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u500b\u6570\\((x_{i}-1)\\)<\/strong>\u306e\u5206\u5e03\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3057\u305f\u3002
\r\n![\"\"](\"https:\/\/since2020.jp\/media\/wp-content\/uploads\/2024\/11\/kasumi_table.png\")
\r\n![\"\"](\"https:\/\/since2020.jp\/media\/wp-content\/uploads\/2024\/11\/kasumi_dist.png\")
\r\n
\r\n<\/p>\r\n\u6a19\u672c\u5e73\u5747\u306e\u8a08\u7b97<\/span><\/strong><\/p>\r\n\u53ce\u96c6\u3057\u305f100\u56de\u5206\u306e\u30c7\u30fc\u30bf\\(x_i\\ (i=1,2,…,N)\\)\u304b\u3089\u8a08\u7b97\u3055\u308c\u308b\\(X\\)\u306e\u6a19\u672c\u5e73\u5747\u306f$$ \\bar{x}=\\frac{\\sum_{i=1}^{100}x_{i}}{100}=1.89 $$\u3067\u3057\u305f\u3002<\/p>\r\n
<\/p>\r\n
<\/p>\r\n
<\/p>\n\n
\u6700\u5c24\u63a8\u5b9a<\/h2>\n
\u4eca\u56de\u306f\u3001\u6700\u5c24\u63a8\u5b9a<\/strong>\u3068\u3044\u3046\u65b9\u6cd5\u306b\u3088\u3063\u3066\u8868\u304c\u51fa\u308b\u78ba\u7387\\(p\\)\u3092\u70b9\u63a8\u5b9a\u3057\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\r\n\u6700\u5c24\u63a8\u5b9a\u306e\u8a73\u7d30\u306a\u8aac\u660e\u306f\u7701\u7565\u3057\u307e\u3059\u3002\u4f8b\u3048\u3070\u4ee5\u4e0b\u306e\u8a18\u4e8b\u304c\u53c2\u8003\u306b\u306a\u308a\u307e\u3059\u3002[blogcard url=”https:\/\/datachemeng.com\/wp-content\/uploads\/2017\/08\/maximumlikelihoodestimation.pdf”]<\/p>\r\n
1: \u5c24\u5ea6\u95a2\u6570\u306e\u5b9a\u5f0f\u5316
\r\n<\/strong><\/span>1\u679a\u306e\u30ab\u30b9\u30df\u304c\u30b3\u30a4\u30f3\u30c8\u30b9\u3092\u884c\u3063\u305f\u56de\u6570\u3092 \\(x_i (i=1,2, \\ldots ,N)\\) \u3068\u8868\u3059\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\r\n\u4eca\u56de\u306f100\u56de\u5206\u306e\u30c7\u30fc\u30bf\u3092\u96c6\u3081\u305f\u306e\u3067\u3001\u30b5\u30f3\u30d7\u30eb\u30b5\u30a4\u30ba\u306f\\(N=100\\)\u3067\u3059\u3002<\/p>\r\n
\u3053\u306e\u3068\u304d\u3001\u5c24\u5ea6\u95a2\u6570\\(L(p)\\)\u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u307e\u3059\u3002<\/p>\r\n
$$ L(p)=\\prod_{i=1}^{N}P(X=x_{i})=\\prod_{i=1}^{N}p^{x_{i}-1}(1-p) $$<\/p>\r\n
\u6700\u5927\u5316\u8a08\u7b97\u3092\u884c\u3044\u3084\u3059\u3044\u3088\u3046\u306b\u5bfe\u6570\u5c24\u5ea6\u95a2\u6570\\(\\mathcal{L}(p)\\)\u3092\u6c42\u3081\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\r\n
$$ \\begin{align*} \\mathcal{L}(p) &=\\log L(p)\\\\ &=\\sum_{i=1}^{N} \\log p^{x_{i}-1}(1-p) \\\\ &=\\sum_{i=1}^{N}\\{ \\log p^{x_{i}-1}+\\log (1-p)\\} \\\\ &=\\sum_{i=1}^{N} \\{ (x_{i}-1)\\log p+\\log (1-p) \\} \\end{align*} $$<\/p>\r\n
<\/p>\r\n
2: \u5c24\u5ea6\u95a2\u6570\u306e\u6700\u5927\u5316
\r\n<\/strong><\/span>\u6700\u5c24\u63a8\u5b9a\u91cf\u3092\u6c42\u3081\u308b\u305f\u3081\u306b\u3001\u5bfe\u6570\u5c24\u5ea6\u95a2\u6570\\(\\mathcal{L}(p)\\)\u3092\\(p\\)\u3067\u5fae\u5206\u3057\u3066\\(=0\\)\u3068\u304a\u304d\u307e\u3059\u3002<\/p>\r\n$$ \\begin{align*} \\frac{d\\mathcal{L}(p)}{dp} &=\\frac{\\sum_{i=1}^{N} \\{ (x_{i}-1)\\log p+\\log (1-p) \\}}{dp}\\\\ &=\\sum_{i=1}^{N} (\\frac{x_{i}-1}{p}-\\frac{1}{1-p})\\\\ &=\\frac{\\sum_{i=1}^{N}x_{i}-N}{p}-\\frac{N}{1-p}\\\\ &=0 \\end{align*} $$<\/p>\r\n
\u3053\u306e\u5f0f\u3092\\(p\\)\u306b\u3064\u3044\u3066\u89e3\u304d\u307e\u3059\u3002<\/p>\r\n
$$ \\begin{split} \\frac{\\sum_{i=1}^{N}x_{i}-N}{p}=\\frac{N}{1-p}\\\\ (1-p)(\\sum_{i=1}^{N}x_{i}-N)=pN\\\\ (1-p)\\sum_{i=1}^{N}x_{i}-(1-p)N=pN \\\\ \\sum_{i=1}^{N}x_{i}-p\\sum_{i=1}^{N}x_{i}=N\\\\ p\\sum_{i=1}^{N}x_{i}=\\sum_{i=1}^{N}x_{i}-N \\end{split} $$<\/p>\r\n
\u8868\u304c\u51fa\u308b\u78ba\u7387\\(p\\)\u306e\u6700\u5c24\u63a8\u5b9a\u91cf\\(\\hat{p}\\)\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\r\n
$$ \\begin{split} \\hat{p} &=1-\\frac{N}{\\sum_{i=1}^{N}x_{i}}\\\\ &=1-\\frac{1}{\\bar{x}} \\qquad \\Big(\\bar{x}=\\frac{\\sum_{i=1}^{N}x_{i}}{N}\\Big) \\end{split} $$<\/p>\r\n
\u3053\u3053\u307e\u3067\u306e\u5f0f\u5909\u5f62\u304c\u7169\u96d1\u306b\u611f\u3058\u305f\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u304c\u3001\u6700\u7d42\u7684\u306b\u306f\\(p\\)\u306e\u6700\u5c24\u63a8\u5b9a\u91cf\u306f\\(X\\)\u306e\u6a19\u672c\u5e73\u5747\\(\\bar{x}\\)\u3092\u7528\u3044\u305f\u7c21\u5358\u306a\u5f0f\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\r\n
<\/p>\r\n
3: \u53ce\u96c6\u3057\u305f\u30c7\u30fc\u30bf\u306e\u4ee3\u5165
\r\n<\/strong><\/span>\u3044\u3088\u3044\u3088\u672c\u984c\u3067\u3059\u3002\u6700\u5c24\u63a8\u5b9a\u91cf\\(\\hat{p}=1-1\/\\bar{x}\\)\u306b\\(N=100\\)\u3068\u53ce\u96c6\u3057\u305f\u30c7\u30fc\u30bf\\(x_{1},x_{2}, \\ldots ,x_{100}\\)\u3092\u4ee3\u5165\u3057\u3066\u3001\u30ab\u30b9\u30df\u304c\u30b3\u30a4\u30f3\u30c8\u30b9\u3067\u8868\u3092\u51fa\u3059\u78ba\u7387\\(p\\)\u306e\u6700\u5c24\u63a8\u5b9a\u5024\u3092\u6c42\u3081\u307e\u3059\u3002<\/p>\r\n\u53ce\u96c6\u3057\u305f\u30c7\u30fc\u30bf\u304b\u3089\u5f97\u3089\u308c\u305f\u6a19\u672c\u5e73\u5747\u306f\\(\\bar{x}=1.89\\)\u3060\u3063\u305f\u306e\u3067\u3001\\(p\\)\u306e\u6700\u5c24\u63a8\u5b9a\u5024\u306f<\/p>\r\n
$$ \\begin{split} \\hat{p} = 1-\\frac{1}{1.89} \\approx 0.471 \\end{split} $$<\/p>\r\n
\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\r\n
\u4eca\u56de\u53ce\u96c6\u3057\u305f\u30c7\u30fc\u30bf\u304b\u3089\u306f\u3001\u30ab\u30b9\u30df\u304c\u30b3\u30a4\u30f3\u306e\u8868\u3092\u51fa\u3059\u78ba\u7387\u306f47.1%\u3067\u3042\u308b<\/strong>\u3068\u63a8\u5b9a\u3055\u308c\u307e\u3057\u305f\uff01<\/p>\r\n<\/p>\n\n
\u307e\u3068\u3081<\/h2>\n
\u672c\u30d6\u30ed\u30b0\u8a18\u4e8b\u3067\u306f\u3001\u5e7e\u4f55\u5206\u5e03\u3092\u5229\u7528\u3057\u3066\u30ab\u30b9\u30df\u306e\u30b3\u30a4\u30f3\u30c8\u30b9\u3092\u78ba\u7387\u30e2\u30c7\u30eb\u3067\u8868\u73fe\u3057\u307e\u3057\u305f\u3002\u3055\u3089\u306b\u6700\u5c24\u63a8\u5b9a\u91cf\u3092\u5c0e\u51fa\u3057\u3001\u5b9f\u969b\u306b\u30ab\u30b9\u30df\u3092\u4f7f\u3063\u3066\u53ce\u96c6\u3057\u305f\u30c7\u30fc\u30bf\u3092\u4ee3\u5165\u3059\u308b\u3053\u3068\u3067\u8868\u304c\u51fa\u308b\u78ba\u7387\\(p\\)\u3092\u70b9\u63a8\u5b9a\u3057\u307e\u3057\u305f\u3002<\/p>\r\n
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\u305d\u3053\u3067\u3001\u6b21\u56de\u306e\u30d6\u30ed\u30b0\u8a18\u4e8b\u3067\u306f\u3001\u7d71\u8a08\u7684\u4eee\u8aac\u691c\u5b9a<\/strong>\u3092\u7528\u3044\u3066\u300c\u30ab\u30b9\u30df\u304c\u30b3\u30a4\u30f3\u306e\u8868\u3092\u51fa\u3059\u78ba\u7387\u306f1\/2\u3088\u308a\u3082\u4f4e\u3044\u3068\u3044\u3048\u308b\u306e\u304b\uff1f\u300d\u3092\u691c\u8a3c\u3057\u307e\u3059\uff01<\/p>\r\n <\/p>\r\n
\u8ffd\u8a18 (2024\/11\/29)<\/span><\/strong>
\r\n\u7d9a\u304d\u306e\u8a18\u4e8b\u3092\u516c\u958b\u3057\u307e\u3057\u305f\u3002[blogcard url=”https:\/\/blog.since2020.jp\/data_analysis\/poke-kasumi-testing\/”]<\/p>\r\n<\/p>\n\n
\u304a\u307e\u3051<\/h2>\n
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\r\n![\"\"](\"https:\/\/since2020.jp\/media\/wp-content\/uploads\/2024\/11\/coin_result.jpg\")
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\r\n![\"\"](\"https:\/\/since2020.jp\/media\/wp-content\/uploads\/2024\/11\/coin_max.png\")
<\/p>\n\n
\u53c2\u8003<\/h2>\n
\u30dd\u30b1\u30dd\u30b1\u516c\u5f0f\u30b5\u30a4\u30c8[blogcard url=”https:\/\/www.pokemontcgpocket.com\/ja\/”]<\/p>","protected":false},"excerpt":{"rendered":"
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![\"\"](\"https:\/\/since2020.jp\/media\/wp-content\/uploads\/2024\/11\/kasumi-1.png\")
<\/p>\r\n