{"id":19904,"date":"2025-04-29T17:00:26","date_gmt":"2025-04-29T09:00:26","guid":{"rendered":"https:\/\/blog.iyatt.com\/?p=19904"},"modified":"2025-04-30T08:51:54","modified_gmt":"2025-04-30T00:51:54","slug":"%e6%9c%9f%e6%9c%9b-expected-value%e3%80%90%e7%bc%96%e8%be%91%e4%b8%ad%e3%80%91","status":"publish","type":"post","link":"https:\/\/blog.iyatt.com\/?p=19904","title":{"rendered":"\u671f\u671b Expected Value"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 ez-toc-wrap-center counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title ez-toc-toggle\" style=\"cursor:pointer\">\u76ee\u5f55<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/blog.iyatt.com\/?p=19904\/#%E6%A6%82%E5%BF%B5\" >\u6982\u5ff5<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/blog.iyatt.com\/?p=19904\/#%E7%A6%BB%E6%95%A3%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F\" >\u79bb\u6563\u968f\u673a\u53d8\u91cf<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/blog.iyatt.com\/?p=19904\/#%E8%BF%9E%E7%BB%AD%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F\" >\u8fde\u7eed\u968f\u673a\u53d8\u91cf<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/blog.iyatt.com\/?p=19904\/#%E4%B8%BE%E4%BE%8B\" >\u4e3e\u4f8b<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/blog.iyatt.com\/?p=19904\/#%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83%E7%9A%84%E6%9C%9F%E6%9C%9B\" >\u6b63\u6001\u5206\u5e03\u7684\u671f\u671b<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h1><span class=\"ez-toc-section\" id=\"%E6%A6%82%E5%BF%B5\"><\/span>\u6982\u5ff5<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>\u671f\u671b\u503c\u53ef\u4ee5\u7406\u89e3\u4e3a\u201c\u5e73\u5747\u7ed3\u679c\u201d\u3002\u65e0\u8bba\u968f\u673a\u53d8\u91cf\u662f\u79bb\u6563\u7684\u8fd8\u662f\u8fde\u7eed\u7684\uff0c\u671f\u671b\u503c\u90fd\u53cd\u6620\u4e86\u968f\u673a\u53d8\u91cf\u53d6\u503c\u7684\u201c\u4e2d\u5fc3\u4f4d\u7f6e\u201d\u3002<\/p>\n<p>\u5927\u6570\u5b9a\u5f8b\uff08Law of large numbers\uff09\uff1a\u662f\u63cf\u8ff0\u76f8\u5f53\u591a\u6b21\u6570\u91cd\u590d\u5b9e\u9a8c\u7684\u7ed3\u679c\u7684\u5b9a\u5f8b\u3002\u6839\u636e\u8fd9\u4e2a\u5b9a\u5f8b\u77e5\u9053\uff0c\u6837\u672c\u6570\u91cf\u8d8a\u591a\uff0c\u5219\u5176\u7b97\u672f\u5e73\u5747\u503c\u5c31\u6709\u8d8a\u9ad8\u7684\u6982\u7387\u63a5\u8fd1\u671f\u671b\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E7%A6%BB%E6%95%A3%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F\"><\/span>\u79bb\u6563\u968f\u673a\u53d8\u91cf<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u5bf9\u4e8e\u79bb\u6563\u968f\u673a\u53d8\u91cf <code class=\"katex-inline\">\\mathbf X<\/code>\uff0c\u5176\u671f\u671b<code class=\"katex-inline\">\\mathbf E(\\mathbf X)=\\sum_{i=1}^{n}x_iP(x_i)<\/code><\/p>\n<p>\u5176\u4e2d\uff0c<code class=\"katex-inline\">x_i<\/code> \u662f\u968f\u673a\u53d8\u91cf <code class=\"katex-inline\">\\mathbf X<\/code> \u7684\u53ef\u80fd\u53d6\u503c\uff0c<code class=\"katex-inline\">P(x_i)<\/code>\u662f<code class=\"katex-inline\">x_i<\/code>\u51fa\u73b0\u7684\u6982\u7387<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E8%BF%9E%E7%BB%AD%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F\"><\/span>\u8fde\u7eed\u968f\u673a\u53d8\u91cf<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u5bf9\u4e8e\u8fde\u7eed\u968f\u673a\u53d8\u91cf <code class=\"katex-inline\">\\mathbf X<\/code>\uff0c\u5176\u671f\u671b<code class=\"katex-inline\">\\mathbf E(\\mathbf X)=\\int_{-\\infty}^{+\\infty}x\\cdot f(x)dx<\/code><\/p>\n<p>\u5176\u4e2d\uff0cf(x)\u662f<code class=\"katex-inline\">\\mathbf X<\/code>\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570<\/p>\n<h1><span class=\"ez-toc-section\" id=\"%E4%B8%BE%E4%BE%8B\"><\/span>\u4e3e\u4f8b<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<h2><span class=\"ez-toc-section\" id=\"%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83%E7%9A%84%E6%9C%9F%E6%9C%9B\"><\/span>\u6b63\u6001\u5206\u5e03\u7684\u671f\u671b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u6b63\u6001\u5206\u5e03\u57fa\u672c\u6982\u5ff5\u53c2\u8003\uff1a<a href=\"https:\/\/blog.iyatt.com\/?p=19840\">https:\/\/blog.iyatt.com\/?p=19840<\/a><\/p>\n<p>\u6b63\u6001\u5206\u5e03\u7684\u671f\u671b\u4e3a<br \/>\n<code class=\"katex-inline\">E(X)=\\int_{-\\infty}^{+\\infty}x\\cdot\\frac{1}{\\sigma\\sqrt{2\\pi}}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}dx<\/code><\/p>\n<p>\u4ee4<code class=\"katex-inline\">u=\\frac{x-\\mu}{\\sigma}<\/code><\/p>\n<p>\u5219<code class=\"katex-inline\">x=\\mu+u\\sigma\uff0cdx=\\sigma du<\/code><\/p>\n<pre><code class=\"language-katex\">\\begin{aligned}\nE(x)&amp;=\\int_{-\\infty}^{+\\infty}(\\mu+u\\sigma)\\frac{1}{\\sigma\\sqrt{2\\pi}}e^{-\\frac{u^2}{2}}\\sigma du \\\\\n&amp;=\\int_{-\\infty}^{+\\infty}(\\mu+u\\sigma)\\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{u^2}{2}} du\\\\\n&amp;=\\mu\\int_{\\infty}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{u^2}{2}}du+\\sigma\\int_{-\\infty}^{+\\infty}\\frac{u}{\\sqrt{2\\pi}}e^{-\\frac{u^2}{2}}du \\\\\n&amp;\\xlongequal[\u7b2c\u4e8c\u90e8\u5206\u88ab\u79ef\u5206\u51fd\u6570\u662f\u5947\u51fd\u6570\uff0c\u79ef\u5206\u7ed3\u679c\u4e3a0]{\u7b2c1\u90e8\u5206\u88ab\u79ef\u5206\u51fd\u6570\u4e3a\u6807\u51c6\u6b63\u6001\u5206\u5e03\uff0c\u79ef\u5206\u7ed3\u679c\u4e3a1}\\mu\\cdot1+\\sigma\\cdot0 \\\\\n&amp;=\\mu\n\\end{aligned}<\/code><\/pre>\n<p>\u5373\u6b63\u6001\u5206\u5e03\u7684\u671f\u671b<code class=\"katex-inline\">E(x)=\\mu<\/code><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6982\u5ff5 \u671f\u671b\u503c\u53ef\u4ee5\u7406\u89e3\u4e3a\u201c\u5e73\u5747\u7ed3\u679c\u201d\u3002\u65e0\u8bba\u968f\u673a\u53d8\u91cf\u662f\u79bb\u6563\u7684\u8fd8\u662f\u8fde\u7eed\u7684\uff0c\u671f\u671b\u503c\u90fd\u53cd\u6620\u4e86\u968f\u673a\u53d8\u91cf\u53d6\u503c\u7684\u201c\u4e2d\u5fc3\u4f4d\u7f6e\u201d\u3002 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"zakra_page_container_layout":"customizer","zakra_page_sidebar_layout":"customizer","zakra_remove_content_margin":false,"zakra_sidebar":"customizer","zakra_transparent_header":"customizer","zakra_logo":0,"zakra_main_header_style":"default","zakra_menu_item_color":"","zakra_menu_item_hover_color":"","zakra_menu_item_active_color":"","zakra_menu_active_style":"","zakra_page_header":true,"_lmt_disableupdate":"no","_lmt_disable":"no","footnotes":""},"categories":[1,612],"tags":[],"class_list":["post-19904","post","type-post","status-publish","format-standard","hentry","category-all","category-612"],"modified_by":"IYATT-yx","_links":{"self":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/19904","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=19904"}],"version-history":[{"count":8,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/19904\/revisions"}],"predecessor-version":[{"id":19914,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/19904\/revisions\/19914"}],"wp:attachment":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=19904"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=19904"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=19904"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}