{"id":19840,"date":"2025-04-07T20:26:25","date_gmt":"2025-04-07T12:26:25","guid":{"rendered":"https:\/\/blog.iyatt.com\/?p=19840"},"modified":"2025-04-30T08:27:55","modified_gmt":"2025-04-30T00:27:55","slug":"%e6%ad%a3%e6%80%81%e5%88%86%e5%b8%83-normal-distribution","status":"publish","type":"post","link":"https:\/\/blog.iyatt.com\/?p=19840","title":{"rendered":"\u6b63\u6001\u5206\u5e03 Normal Distribution"},"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=19840\/#%E6%A6%82%E5%BF%B5\" >\u6982\u5ff5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/blog.iyatt.com\/?p=19840\/#%E8%AE%A1%E7%AE%97%E6%9C%BA%E8%AE%A1%E7%AE%97%E5%92%8C%E7%BB%98%E5%9B%BE\" >\u8ba1\u7b97\u673a\u8ba1\u7b97\u548c\u7ed8\u56fe<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/blog.iyatt.com\/?p=19840\/#%E6%B5%8B%E8%AF%95%E7%8E%AF%E5%A2%83\" >\u6d4b\u8bd5\u73af\u5883<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/blog.iyatt.com\/?p=19840\/#%E5%B7%B2%E7%9F%A5%E5%9D%87%E5%80%BC%E3%80%81%E6%A0%87%E5%87%86%E5%B7%AE%E7%BB%98%E5%9B%BE\" >\u5df2\u77e5\u5747\u503c\u3001\u6807\u51c6\u5dee\u7ed8\u56fe<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/blog.iyatt.com\/?p=19840\/#%E4%B8%89%E4%B8%AA%E5%85%B8%E5%9E%8B%E5%8C%BA%E9%97%B4%E6%A6%82%E7%8E%87\" >\u4e09\u4e2a\u5178\u578b\u533a\u95f4\u6982\u7387<\/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>\u82e5\u968f\u673a\u53d8\u91cf<code class=\"katex-inline\">\\mathbf{X}<\/code>\u670d\u4ece\u4e00\u4e2a\u5e73\u5747\u6570\u4e3a<code class=\"katex-inline\">\\mu<\/code>\u3001\u6807\u51c6\u5dee\u4e3a<code class=\"katex-inline\">\\sigma<\/code>\u7684\u6b63\u6001\u5206\u5e03\uff0c\u5219\u8bb0\u4e3a\uff1a<br \/>\n<code class=\"katex-inline\">\\mathbf{X\\sim N(\\mu,\\sigma^2)}<\/code><\/p>\n<p>\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\uff08Probability Density Function\uff09\uff1a<code class=\"katex-inline\">f(x)=\\frac{1}{\\sigma\\sqrt{2\\pi}}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}<\/code><\/p>\n<p>\u5f53<code class=\"katex-inline\">\\mu=0<\/code>\uff0c<code class=\"katex-inline\">\\sigma=1<\/code>\u65f6\u4e3a\u6807\u51c6\u6b63\u6001\u5206\u5e03\uff0c\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4e3a<code class=\"katex-inline\">f(x)=\\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{x^2}{2}}<\/code><\/p>\n<h1><span class=\"ez-toc-section\" id=\"%E8%AE%A1%E7%AE%97%E6%9C%BA%E8%AE%A1%E7%AE%97%E5%92%8C%E7%BB%98%E5%9B%BE\"><\/span>\u8ba1\u7b97\u673a\u8ba1\u7b97\u548c\u7ed8\u56fe<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<h2><span class=\"ez-toc-section\" id=\"%E6%B5%8B%E8%AF%95%E7%8E%AF%E5%A2%83\"><\/span>\u6d4b\u8bd5\u73af\u5883<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li>Python 3.13.1<\/li>\n<li>Jupyter==1.11.1<\/li>\n<li>matplotlib==3.10.1<\/li>\n<li>numpy==2.2.4<\/li>\n<li>sympy==1.13.3<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"%E5%B7%B2%E7%9F%A5%E5%9D%87%E5%80%BC%E3%80%81%E6%A0%87%E5%87%86%E5%B7%AE%E7%BB%98%E5%9B%BE\"><\/span>\u5df2\u77e5\u5747\u503c\u3001\u6807\u51c6\u5dee\u7ed8\u56fe<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Jupyter<\/p>\n<pre><code class=\"language-python\">import numpy as np\nimport matplotlib.pyplot as plt\n\n# \u5747\u503c\nmu = 0\n# \u6807\u51c6\u5dee\nsigma = 1\n\n# \u751f\u6210 X \u503c\n# \u751f\u6210 100 \u4e2a\u70b9\uff0c\u8303\u56f4\u4ece \u03bc-3\u03c3 \u5230 \u03bc+3\u03c3\nx = np.linspace(mu - 3 * sigma, mu + 3 * sigma, 100)\n\n# \u8ba1\u7b97 x \u503c\u5bf9\u5e94\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u503c\ny = np.e ** (-(x - mu) ** 2 \/ (2 * sigma ** 2)) \/ (sigma * np.sqrt(2 * np.pi))\n\n# \u7ed8\u5236\u6b63\u6001\u5206\u5e03\u66f2\u7ebf\nplt.plot(x, y, label=&#039;Normal Distribution&#039;)\n\n# \u6dfb\u52a0\u56fe\u4f8b\nplt.legend()\n\n# \u6dfb\u52a0\u5750\u6807\u8f74\u6807\u7b7e\nplt.xlabel(&#039;x&#039;)\nplt.ylabel(&#039;Probability Density&#039;)\n\n# \u663e\u793a\u56fe\u5f62\nplt.show()<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2025\/04\/image-1744028774079.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 576px; --smush-placeholder-aspect-ratio: 576\/432;\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E4%B8%89%E4%B8%AA%E5%85%B8%E5%9E%8B%E5%8C%BA%E9%97%B4%E6%A6%82%E7%8E%87\"><\/span>\u4e09\u4e2a\u5178\u578b\u533a\u95f4\u6982\u7387<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Jupyter<\/p>\n<pre><code class=\"language-python\">import sympy as sy\n\n# \u5b9a\u4e49\u7b26\u53f7\nx, mu = sy.symbols(&#039;x mu&#039;, real=True) # \u5b9e\u6570\nsigma = sy.symbols(&#039;sigma&#039;, real=True, positive=True) # \u6b63\u5b9e\u6570\n# \u5b9a\u4e49\u6b63\u6001\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\nf = sy.E ** (-(x - mu) ** 2 \/ (2 * sigma ** 2)) \/ (sigma * sy.sqrt(2 * sy.pi))\n\n# \u8ba1\u7b97 -\u221e \u5230 +\u221e \u7684\u79ef\u5206\nall =sy.integrate(f, (x, -sy.oo, sy.oo))\nprint(&#039;-\u221e \u5230 +\u221e \u7684\u79ef\u5206=&#039;)\ndisplay(all)\n\n# \u8ba1\u7b97 \u03bc-\u03c3 \u5230 \u03bc+\u03c3 \u7684\u79ef\u5206\nresult1 = sy.integrate(f, (x, mu - 1 * sigma, mu + 1 * sigma))\nprint(&#039;\u03bc-\u03c3 \u5230 \u03bc+\u03c3 \u7684\u79ef\u5206=&#039;)\ndisplay(result1)\nprint(result1.evalf(16))\n\n# \u8ba1\u7b97 \u03bc-2\u03c3 \u5230 \u03bc+2\u03c3 \u7684\u79ef\u5206\nresult2 = sy.integrate(f, (x, mu - 2 * sigma, mu + 2 * sigma))\nprint(&#039;\u03bc-2\u03c3 \u5230 \u03bc+2\u03c3 \u7684\u79ef\u5206=&#039;)\ndisplay(result2)\nprint(result2.evalf(16))\n\n# \u8ba1\u7b97 \u03bc-\u03c3 \u5230 \u03bc+\u03c3 \u7684\u79ef\u5206\nresult3 = sy.integrate(f, (x, mu - 3 * sigma, mu + 3 * sigma))\nprint(&#039;\u03bc-3\u03c3 \u5230 \u03bc+3\u03c3 \u7684\u79ef\u5206=&#039;)\ndisplay(result3)\nprint(result3.evalf(16))<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2025\/04\/image-1744114378208.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 209px; --smush-placeholder-aspect-ratio: 209\/436;\" \/><\/p>\n<p>\u5373<br \/>\n<code class=\"katex-inline\">P(\\mu\\pm\\sigma)=erf(\\frac{\\sqrt{2}}{2})\\approx0.6826894921370859<\/code><br \/>\n<code class=\"katex-inline\">P(\\mu\\pm2\\sigma)=erf(\\sqrt{2})\\approx0.9544997361036416<\/code><br \/>\n<code class=\"katex-inline\">P(\\mu\\pm3\\sigma)=erf(\\frac{3\\sqrt{2}}{2})\\approx0.9973002039367398<\/code><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6982\u5ff5 \u82e5\u968f\u673a\u53d8\u91cf\\mathbf{X}\u670d\u4ece\u4e00\u4e2a\u5e73\u5747\u6570\u4e3a\\mu\u3001\u6807\u51c6\u5dee\u4e3a\\sigma\u7684\u6b63\u6001\u5206\u5e03\uff0c\u5219\u8bb0\u4e3a\uff1a \\mat [&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":[1,606,592,612],"tags":[],"class_list":["post-19840","post","type-post","status-publish","format-standard","hentry","category-all","category-jupyter","category-python","category-612"],"modified_by":"IYATT-yx","_links":{"self":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/19840","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=19840"}],"version-history":[{"count":15,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/19840\/revisions"}],"predecessor-version":[{"id":19910,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/19840\/revisions\/19910"}],"wp:attachment":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=19840"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=19840"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=19840"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}