{"id":1104,"date":"2020-06-15T11:00:00","date_gmt":"2020-06-15T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=1104"},"modified":"2020-06-13T10:27:37","modified_gmt":"2020-06-13T01:27:37","slug":"conditional-variance","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/conditional-variance","title":{"rendered":"\u6761\u4ef6\u4ed8\u304d\u5206\u6563\u3010\u5b9a\u7fa9\u304b\u3089\u5fb9\u5e95\u89e3\u8aac\uff01\u3011"},"content":{"rendered":"\n

\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2606\u2606\u2606\u2606<\/span><\/p>\n\n\n\n\n\n\n\n

\u78ba\u7387\u5909\u6570\u306e\u7d44\\((X,\\ Y)\\)\u304c\u5f97\u3089\u308c\u305f\u3068\u304d\u3092\u8003\u3048\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u78ba\u7387\u5909\u6570\\(Y\\)\u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u4e0b\u3067\u3001\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u5206\u6563\u3092\u8003\u3048\u307e\u3059\u3002\u3053\u306e\u5206\u6563\u3092\u6761\u4ef6\u4ed8\u304d\u5206\u6563\u3068\u547c\u3073\u307e\u3059\u3002\u6761\u4ef6\u4ed8\u304d\u5206\u6563\u3092\u7406\u89e3\u3059\u308b\u306b\u306f\u3001\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u306e\u7406\u89e3\u3068\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024\u306e\u7406\u89e3\u304c\u5fc5\u9808\u306a\u306e\u3067\u5fa9\u7fd2\u3057\u3066\u304a\u304f\u3053\u3068\u3092\u30aa\u30b9\u30b9\u30e1\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u203b\u3000\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u306b\u3064\u3044\u3066\u306f\uff1c\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u306e\u8a18\u4e8b\uff1e<\/a>\u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

\u203b\u3000\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024\u306b\u3064\u3044\u3066\u306f\uff1c\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024\u306e\u8a18\u4e8b\uff1e<\/a>\u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

\u6761\u4ef6\u4ed8\u304d\u5206\u6563\u306e\u5b9a\u7fa9<\/h3>\n\n\n\n
\u6761\u4ef6\u4ed8\u304d\u5206\u6563(conditional variance)<\/strong><\/div>\n
\u78ba\u7387\u5909\u6570\u306e\u7d44\\((X,\\ Y)\\)\u304c\u5f97\u3089\u308c\u305f\u3068\u304d\u3001\u78ba\u7387\u5909\u6570\\(Y\\)\u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u4e0b\u3067\u3001\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u5206\u6563\u3092\u3001\u6761\u4ef6\u4ed8\u304d\u5206\u6563\u3068\u547c\u3073\\(\\mathrm{Var}[X|Y]\\)\u3067\u8868\u3057\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002\\[\n\\mathrm{Var}[X|Y] = \\left\\{
\\begin{align}\n&\\sum_{i}(x_{i}-\\mathrm{E}[X|Y])^{2}f(x_{i}|y)\\ \\ \\ \uff08\u96e2\u6563\u578b\uff09\\\\\n&\\int_{-\\infty}^{\\infty}(x-\\mathrm{E}[X|Y])^{2}f(x|y)dx\\ \\ \\ \uff08\u9023\u7d9a\u578b\uff09\n\\end{align}<\/div>\\right.\n\\]\u3053\u3053\u3067\u3001\\(f(x_{i}|y)\\)\u306f\\(Y\\)\u3092\u4e0e\u3048\u305f\u4e0b\u3067\u306e\\(X\\)\u306e\u78ba\u7387\u95a2\u6570\u3092\u8868\u3057\u3001\\(f(x|y)\\)\u306f\\(Y\\)\u3092\u4e0e\u3048\u305f\u4e0b\u3067\u306e\\(X\\)\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u8868\u3057\u3066\u3044\u307e\u3059\u3002<\/div>\n

 <\/p>\n\n\n\n

<\/div>

\u6761\u4ef6\u4ed8\u304d\u5206\u6563\u304c\u610f\u5473\u3059\u308b\u3053\u3068\u306f\u3001\u666e\u901a\u306e\u5206\u6563\u3068\u5168\u304f\u540c\u3058\u3067\u3059\u3002<\/p>

<\/div><\/div><\/div>\n\n\n\n

\u6761\u4ef6\u4ed8\u304d\u5206\u6563\u306f\u4ee5\u4e0b\u306e\u5f0f\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u3053\u3068\u304c\u3042\u308a\u307e\u3059\u304c\u3001\u4e0a\u306e\u5b9a\u7fa9\u3068\u540c\u3058\u306a\u306e\u3067\u3069\u3061\u3089\u3067\u899a\u3048\u3066\u3082\u554f\u984c\u3042\u308a\u307e\u305b\u3093\u3002$$\\mathrm{Var}[X|Y]=\\mathrm{E}[X^{2}|Y]-\\mathrm{E}[X|Y]^{2}$$<\/p>\n\n\n\n

\u6761\u4ef6\u4ed8\u304d\u5206\u6563\u306e\u6027\u8cea<\/h3>\n\n\n\n
\u6761\u4ef6\u4ed8\u304d\u5206\u6563\u306e\u6027\u8cea<\/strong><\/div>\n
\u78ba\u7387\u5909\u6570\u306e\u7d44\\((X,\\ Y)\\)\u304c\u5f97\u3089\u308c\u305f\u3068\u304d\u3001\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u5206\u6563\u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002$$\\mathrm{Var}[X] = \\mathrm{E}\\left[ \\mathrm{Var}\\left[ X|Y \\right] \\right]+\\mathrm{Var}\\left[ \\mathrm{E}\\left[ X|Y \\right] \\right]$$<\/div>\n

\u00a0<\/p>\n\n\n\n

\u3053\u306e\u6027\u8cea\u304c\u6210\u308a\u7acb\u3064\u304b\u3069\u3046\u304b\u4e0b\u306b\u8a3c\u660e\u3092\u4e0e\u3048\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\u25a0\u8a3c\u660e<\/p>\n\n\n\n

\u53f3\u8fba\u3092\u5909\u5f62\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u307e\u305a\u3001\u6761\u4ef6\u4ed8\u304d\u5206\u6563\u306e\u5b9a\u7fa9\u304b\u3089

\\begin{align}
\\mathrm{E}\\left[ \\mathrm{Var}\\left[ X|Y \\right] \\right] &= \\mathrm{E}\\left[ \\mathrm{E}[X^{2}|Y]-\\mathrm{E}[X|Y]^{2} \\right] \\\\
&= \\mathrm{E}[X^{2}]-\\mathrm{E}\\left[ \\mathrm{E}[X|Y]^{2} \\right]
\\end{align}<\/div>
\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u3055\u3089\u306b\u3001\u5206\u6563\u306e\u5b9a\u7fa9\u304b\u3089
\\begin{align}
\\mathrm{Var}\\left[ \\mathrm{E}\\left[ X|Y \\right] \\right] &= \\mathrm{E}\\left[ \\mathrm{E}\\left[ X|Y \\right]^{2} \\right]-\\mathrm{E}\\left[ \\mathrm{E}\\left[ X|Y \\right] \\right]^{2} \\\\
&=\\mathrm{E}\\left[ \\mathrm{E}\\left[ X|Y \\right]^{2} \\right]-\\mathrm{E}\\left[ X \\right]^{2}
\\end{align}<\/div>
\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u3053\u306e2\u5f0f\u3092\u7528\u3044\u308b\u3068\u6c42\u3081\u305f\u3044\u5f0f\u304c\u51fa\u3066\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u25a1<\/p>\n\n\n\n

\u6761\u4ef6\u4ed8\u304d\u5206\u6563\u306e\u95a2\u9023\u8a18\u4e8b<\/h3>\n\n\n\n

\u30fb\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024<\/a>\uff1a\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u5909\u6570\u306e\u671f\u5f85\u5024\u306e\u6c42\u3081\u65b9\u3092\u7d39\u4ecb\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\u30fb\u5206\u6563\u306e\u5b9a\u7fa9<\/a>\uff1a\u5206\u6563\u3092\u5b9a\u7fa9\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\u30fb\u5171\u5206\u6563<\/a>\uff1a\u5171\u5206\u6563\u3092\u5b9a\u7fa9\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\u30fb\u5206\u6563\u306e\u6027\u8cea<\/a>\uff1a\u5206\u6563\u306e\u6301\u3064\u6027\u8cea\u3092\u307e\u3068\u3081\u3066\u3044\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

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