{"id":1058,"date":"2020-06-12T11:00:00","date_gmt":"2020-06-12T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=1058"},"modified":"2020-06-13T10:33:33","modified_gmt":"2020-06-13T01:33:33","slug":"conditional-mean","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/conditional-mean","title":{"rendered":"\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024\u3010\u5b9a\u7fa9\u304b\u3089\u5fb9\u5e95\u89e3\u8aac\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\u671f\u5f85\u5024\u3092\u8003\u3048\u307e\u3059\u3002\u3053\u306e\u671f\u5f85\u5024\u3092\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024\u3068\u547c\u3073\u307e\u3059\u3002\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024\u3092\u7406\u89e3\u3059\u308b\u306b\u306f\u3001\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\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

\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024\u306e\u5b9a\u7fa9<\/h3>\n\n\n\n
\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024(conditional mean)<\/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\u671f\u5f85\u5024\u3092\u3001\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024\u3068\u547c\u3073\\(\\mathrm{E}[X|Y]\\)\u3067\u8868\u3057\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002\\[\n\\mathrm{E}[X|Y] = \\left\\{
\\begin{align}\n&\\sum_{i}x_{i}f(x_{i}|y)\\ \\ \\ \uff08\u96e2\u6563\u578b\uff09\\\\\n&\\int_{-\\infty}^{\\infty}xf(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\u671f\u5f85\u5024\u304c\u610f\u5473\u3059\u308b\u3053\u3068\u306f\u3001\u666e\u901a\u306e\u671f\u5f85\u5024\u3068\u5168\u304f\u540c\u3058\u3067\u3059\u3002<\/p>

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

\u203b\u3000\u671f\u5f85\u5024\u306b\u3064\u3044\u3066\u306f\uff1c\u671f\u5f85\u5024\u306e\u5b9a\u7fa9\u306e\u8a18\u4e8b\uff1e<\/a>\u3092\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024\u306e\u6027\u8cea<\/h3>\n\n\n\n
\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024\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\\(Y\\)\u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u4e0b\u3067\u3001\u78ba\u7387\u5909\u6570\\(X\\)\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024\\(\\mathrm{E}[X|Y]\\)\u306b\u306f\u6b21\u306e\u6027\u8cea\u304c\u3042\u308a\u307e\u3059\u3002$$\\mathrm{E}[X] = \\mathrm{E}_{Y}\\left[ \\mathrm{E}\\left[ X|Y \\right] \\right]$$\u305f\u3060\u3057\u3001\\(\\mathrm{E}_{Y}\\)\u306f\\(Y\\)\u306e\u671f\u5f85\u5024\u3092\u8868\u3057\u307e\u3059\u3002<\/div>\n

 <\/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\u96e2\u6563\u578b\u306e\u307f\u3092\u7d39\u4ecb\u3057\u307e\u3059\u304c\u3001\u9023\u7d9a\u578b\u3067\u3082\u540c\u69d8\u306e\u624b\u9806\u3067\u884c\u3044\u307e\u3059\u3002

\\begin{align}
\\mathrm{E}_{Y}\\left[ \\mathrm{E}\\left[ X|Y \\right] \\right] &= \\mathrm{E}_{Y}\\left[ \\sum_{i}x_{i}f(x_{i}|y) \\right] \\\\
&= \\mathrm{E}_{Y}\\left[ \\sum_{i}x_{i}\\frac{\\mathrm{P}(X=x_{i},\\ Y=y)}{\\mathrm{P}(Y=y)} \\right] \\\\
&= \\sum_{i}\\sum_{j}x_{i}\\frac{\\mathrm{P}(X=x_{i},\\ Y=y_{j})}{\\mathrm{P}(Y=y_{j})}\\mathrm{P}(Y=y_{j}) \\\\
&= \\sum_{i}\\sum_{j}x_{i}\\mathrm{P}(X=x_{i},\\ Y=y_{j}) \\\\
&= \\sum_{i}x_{i}\\mathrm{P}(X=x_{i})\\\\
&= \\mathrm{E}[X]
\\end{align}<\/div><\/p>\n\n\n\n

\u3068\u306a\u308b\u306e\u3067\u3001\u6c42\u3081\u305f\u3044\u5f0f\u304c\u51fa\u3066\u304d\u307e\u3057\u305f\u3002\u3053\u3053\u3067\u5f0f\u5909\u5f62\u3067\u306f\u540c\u6642\u78ba\u7387\u5206\u5e03\u306e\u6027\u8cea\u3092\u5229\u7528\u3057\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

\u203b\u3000\u540c\u6642\u78ba\u7387\u5206\u5e03\u306b\u3064\u3044\u3066\u306f\uff1c\u540c\u6642\u78ba\u7387\u5206\u5e03\u306e\u8a18\u4e8b\uff1e\u3092\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

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

\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024\u306e\u95a2\u9023\u8a18\u4e8b<\/h3>\n\n\n\n

\u30fb\u6761\u4ef6\u4ed8\u304d\u5206\u6563<\/a>\uff1a\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u5909\u6570\u306e\u5206\u6563\u306e\u6c42\u3081\u65b9\u3092\u7d39\u4ecb\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\u30fb\u671f\u5f85\u5024\u306e\u5b9a\u7fa9<\/a>\uff1a\u671f\u5f85\u5024\u3092\u5b9a\u7fa9\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\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\u671f\u5f85\u5024\u306e\u6027\u8cea<\/a>\uff1a\u671f\u5f85\u5024\u306e\u6301\u3064\u6027\u8cea\u3092\u307e\u3068\u3081\u3066\u3044\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2606\u2606\u2606\u2606 \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 … <\/p>\n","protected":false},"author":1,"featured_media":829,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[72],"tags":[73],"jetpack_featured_media_url":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-content\/uploads\/2020\/05\/135-money-images.jpg","_links":{"self":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1058"}],"collection":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/comments?post=1058"}],"version-history":[{"count":14,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1058\/revisions"}],"predecessor-version":[{"id":1160,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/posts\/1058\/revisions\/1160"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media\/829"}],"wp:attachment":[{"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/media?parent=1058"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/categories?post=1058"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/wp-json\/wp\/v2\/tags?post=1058"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}