{"id":14409,"date":"2024-03-31T00:03:23","date_gmt":"2024-03-30T16:03:23","guid":{"rendered":"https:\/\/blog.iyatt.com\/?p=14409"},"modified":"2025-04-07T16:52:02","modified_gmt":"2025-04-07T08:52:02","slug":"%e5%b7%b2%e7%9f%a5%e6%a6%82%e7%8e%87%e5%af%86%e5%ba%a6%e5%87%bd%e6%95%b0%e7%94%9f%e6%88%90%e9%9a%8f%e6%9c%ba%e6%95%b0%e7%9a%84%e6%96%b9%e6%b3%95%ef%bc%88%e7%bc%96%e8%be%91%e4%b8%ad%ef%bc%89","status":"publish","type":"post","link":"https:\/\/blog.iyatt.com\/?p=14409","title":{"rendered":"\u7531\u7ed9\u5b9a\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u751f\u6210\u968f\u673a\u6570\u7684\u65b9\u6cd5"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_84 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=14409\/#1_%E9%80%86%E5%88%86%E5%B8%83%E5%87%BD%E6%95%B0%E6%B3%95%EF%BC%88Inverse_Transform_Method%EF%BC%89\" >1 \u9006\u5206\u5e03\u51fd\u6570\u6cd5\uff08Inverse Transform Method\uff09<\/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=14409\/#%E4%BE%8B%E4%B8%80\" >\u4f8b\u4e00<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/blog.iyatt.com\/?p=14409\/#2_%E8%88%8D%E9%80%89%E6%B3%95%EF%BC%88Acceptance-Rejection_Method%EF%BC%89\" >2 \u820d\u9009\u6cd5\uff08Acceptance-Rejection Method\uff09<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/blog.iyatt.com\/?p=14409\/#%E4%BE%8B%E4%B8%80-2\" >\u4f8b\u4e00<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h1><span class=\"ez-toc-section\" id=\"1_%E9%80%86%E5%88%86%E5%B8%83%E5%87%BD%E6%95%B0%E6%B3%95%EF%BC%88Inverse_Transform_Method%EF%BC%89\"><\/span>1 \u9006\u5206\u5e03\u51fd\u6570\u6cd5\uff08Inverse Transform Method\uff09<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>\u5df2\u77e5\u6982\u7387\u5bc6\u5ea6\u51fd\u6570(Probability Density Function\uff0cPDF)\uff1a f(x)<br \/>\n\u5f97\u5230 f(x) \u7684\u7d2f\u79ef\u5206\u5e03\u51fd\u6570\uff08Cumulative Density Function\uff0cCDF\uff09\uff0c\u4e5f\u79f0\u5206\u5e03\u51fd\u6570\uff1a<code class=\"katex-inline\">F(x)=\\int_{-\\infty}^xf(t)dt<\/code><br \/>\n\u5f97\u5230 F(x) \u7684\u53cd\u51fd\u6570\uff08\u9006\u51fd\u6570\uff09(Inverse Cumulative Density Function\uff0cICDF)\uff1a<code class=\"katex-inline\">x=F^{-1}(y)<\/code><br \/>\n\u751f\u6210\u670d\u4ece U(0,1) \u5747\u5300\u5206\u5e03\u7684\u968f\u673a\u6570 x\uff08\u5373\u6982\u7387\u53d6\u503c\u5728 0-1 \u8303\u56f4\uff09\uff0c\u4ee3\u5165 ICDF \u8ba1\u7b97\u5f97\u5230\u7684\u503c\u5373\u4e3a\u8981\u6c42\u7684\u968f\u673a\u6570\u3002<\/p>\n<p>\u6709\u4e9b\u60c5\u51b5\u4e0b\u4eba\u5de5\u6c42\u89e3 ICDF \u6bd4\u8f83\u56f0\u96be\uff0c\u53ef\u4ee5\u7528 SymPy \u5e93\u5f97\u5230\u3002<br \/>\n\u79ef\u5206\u53c2\u8003\uff1a<a href=\"https:\/\/blog.iyatt.com\/?p=12906#7_%E7%A7%AF%E5%88%86\">https:\/\/blog.iyatt.com\/?p=12906#7_%E7%A7%AF%E5%88%86<\/a><br \/>\n\u53cd\u51fd\u6570\u53c2\u8003\uff1a<a href=\"https:\/\/blog.iyatt.com\/?p=14396\">https:\/\/blog.iyatt.com\/?p=14396<\/a><br \/>\n\u6bd4\u5982\uff1a<br \/>\n<img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/03\/image-1713193392143.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 1313px; --smush-placeholder-aspect-ratio: 1313\/993;\" \/><\/p>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/03\/image-1713193415697.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 1290px; --smush-placeholder-aspect-ratio: 1290\/911;\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E4%BE%8B%E4%B8%80\"><\/span>\u4f8b\u4e00<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>PDF\uff1a<code class=\"katex-inline\">f(x)=\\sqrt{x},x\\ge0<\/code><br \/>\nCDF\uff1a<code class=\"katex-inline\">F(x)=\\int_0^xf(t)dt=\\frac{2}{3}t^{\\frac{3}{2}}|_0^x=\\frac{2}{3}x^{\\frac{3}{2}}<\/code><br \/>\nICDF\uff1a<code class=\"katex-inline\">F^{-1}(x)=(\\frac{3}{2}x)^{\\frac{2}{3}}<\/code><br \/>\n\u5bf9 ICDF\uff0cx \u4e3a\u670d\u4ece U(0,1) \u5747\u5300\u5206\u5e03\u7684\u968f\u673a\u6570<\/p>\n<pre><code class=\"language-py\">import numpy as np\nimport matplotlib.pyplot as plt\n\ndef icdf(x):\n    return (3 * x \/ 2)**(2 \/ 3)\n\nrandom_list = list()\nnumber = 10000\nfor i in range(number):\n    p = np.random.uniform(0, 1)\n    random_list.append(icdf(p))\n\nplt.hist(random_list, bins=20)<\/code><\/pre>\n<p>\u4e0a\u9762\u4ee3\u7801\u8fdb\u884c\u4e86 10000 \u6b21\u968f\u673a\u6570\u751f\u6210\uff0c\u9996\u5148\u751f\u6210\u5747\u5300\u5206\u5e03\u7684\u968f\u673a\u6570\uff0c\u8303\u56f4\u53d6\u503c\u5728 0-1\uff08\u6982\u7387\u7684\u8303\u56f4\uff09\uff0c\u5c06\u751f\u6210\u7684\u6570\u4ee3\u5165 ICDF \u8ba1\u7b97\u51fa\u7684\u503c\u5c31\u662f\u7b26\u5408 PDF \u7684\u968f\u673a\u6570\u3002\u4e0b\u9762\u7ed8\u5236\u4e86\u751f\u6210\u7684 10000 \u4e2a\u968f\u673a\u6570\u7684\u76f4\u65b9\u56fe\uff0c\u6a2a\u5750\u6807\u662f\u6570\u636e\u7684\u533a\u95f4\uff0c\u7eb5\u5750\u6807\u662f\u5bf9\u5e94\u7684\u9891\u6570\u3002<br \/>\n<img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/03\/image-1711814346860.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 550px; --smush-placeholder-aspect-ratio: 550\/412;\" \/><\/p>\n<p>PDF \u7684\u51fd\u6570\u66f2\u7ebf\u56fe\u50cf<\/p>\n<pre><code class=\"language-py\">import matplotlib.pyplot as plt\nimport numpy as np\n\ndef pdf(x):\n    return  x**(1 \/ 2)\n\nX = np.linspace(0, (3 \/ 2)**(2 \/ 3), 1000)\nY = pdf(X)\nplt.plot(X, Y)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/03\/image-1711870535522.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 540px; --smush-placeholder-aspect-ratio: 540\/402;\" \/><\/p>\n<h1><span class=\"ez-toc-section\" id=\"2_%E8%88%8D%E9%80%89%E6%B3%95%EF%BC%88Acceptance-Rejection_Method%EF%BC%89\"><\/span>2 \u820d\u9009\u6cd5\uff08Acceptance-Rejection Method\uff09<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>\u8fd9\u79cd\u65b9\u6cd5\u7406\u8bba\u4e0a\u53ea\u8981\u77e5\u9053 PDF \u5c31\u80fd\u7528\uff0c\u800c\u6709\u4e9b\u60c5\u51b5\u4e0b\uff0c\u9006\u5206\u5e03\u51fd\u6570\u53ef\u80fd\u4e0d\u5b58\u5728\u6216\u8005\u96be\u6c42\uff0c\u7b2c\u4e00\u79cd\u65b9\u6cd5\u5c31\u4e0d\u90a3\u4e48\u9002\u7528\u3002<br \/>\n\u5df2\u77e5 PDF \u4e3a f(x)\uff0c\u5982\u679c\u8981\u751f\u6210\u7684\u968f\u673a\u6570\u8303\u56f4\u662f (a, b)\uff0c\u90a3\u4e48\u751f\u6210\u670d\u4ece U(a,b) \u5747\u5300\u5206\u5e03\u7684\u968f\u673a\u6570 x\uff0c\u518d\u751f\u6210\u670d\u4ece U(0,M) \u7684\u968f\u673a\u6570 y\uff08M \u4e3af(x) \u5728(a,b)\u533a\u95f4\u7684\u6700\u5927\u53d6\u503c\uff0c\u5b9e\u9645 M \u8fd9\u4e2a\u4f4d\u7f6e\u7684\u53d6\u503c\u53ef\u4ee5\u5927\u4e8e\u7b49\u4e8e M, \u53ea\u662f\u4f1a\u964d\u4f4e\u6548\u7387\uff0c\u589e\u5927\u91cd\u590d\u6267\u884c\u7684\u53ef\u80fd\uff09\u3002\u5f53\u6ee1\u8db3 <code class=\"katex-inline\">y\\le f(x)<\/code>\u65f6\uff0cx \u7684\u503c\u5c31\u662f\u8981\u751f\u6210\u7684\u968f\u673a\u6570\uff0c\u5426\u5219\u91cd\u590d\u6267\u884c\u76f4\u5230\u6ee1\u8db3\u6761\u4ef6\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E4%BE%8B%E4%B8%80-2\"><\/span>\u4f8b\u4e00<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>PDF\uff1a<code class=\"katex-inline\">f(x)=\\sqrt{x},x\\ge0<\/code><br \/>\nCDF\uff1a<code class=\"katex-inline\">F(x)=\\int_0^xf(t)dt=\\frac{2}{3}t^{\\frac{3}{2}}|_0^x=\\frac{2}{3}x^{\\frac{3}{2}}<\/code><\/p>\n<p>\u4ee4 CDF = 1\uff0c\u5373\u6982\u7387\u7684\u6700\u5927\u503c\uff0c\u53ef\u5f97\u5230 x \u6700\u5927\u53d6\u503c<code class=\"katex-inline\">(\\frac{3}{2})^{\\frac{2}{3}}<\/code>\uff0c\u4ee3\u5165 PDF \u5f97\u5230 PDF \u6700\u5927\u53d6\u503c <code class=\"katex-inline\">(\\frac{3}{2})^{\\frac{1}{3}}<\/code>.<br \/>\n\u5219\u751f\u6210\u968f\u673a\u6570 <code class=\"katex-inline\">x=U(0,(\\frac{3}{2})^{\\frac{2}{3}})<\/code>\uff0c<code class=\"katex-inline\">y=U(0,(\\frac{3}{2})^{\\frac{1}{3}})<\/code>\uff0c\u518d\u5224\u65ad y&lt;f(x)\uff0c\u6ee1\u8db3\u5219\u53d6 x \u7684\u503c\u4f5c\u4e3a\u968f\u673a\u6570\uff0c\u4e0d\u6ee1\u8db3\u5c31\u91cd\u590d\u6267\u884c\u3002<\/p>\n<pre><code class=\"language-py\">import numpy as np\nimport matplotlib.pyplot as plt\n\ndef pdf(x):\n    return x**(1 \/ 2)\n\nrandom_list = list()\nnumber = 10000\nfor i in range(number):\n    while True:\n        x = np.random.uniform(0, (3 \/ 2)**(2 \/ 3))\n        y = np.random.uniform(0, (3 \/ 2)**(1 \/ 3))\n        if y &lt;= pdf(x):\n            random_list.append(x)\n            break\n\nplt.hist(random_list, bins=20)<\/code><\/pre>\n<p>\u4e0b\u56fe\u662f\u751f\u6210\u7684 10000 \u4e2a\u968f\u673a\u6570\u7684\u5206\u5e03\u56fe<br \/>\n<img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/03\/image-1711872077258.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 543px; --smush-placeholder-aspect-ratio: 543\/403;\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>1 \u9006\u5206\u5e03\u51fd\u6570\u6cd5\uff08Inverse Transform Method\uff09 \u5df2\u77e5\u6982\u7387\u5bc6\u5ea6\u51fd\u6570(Probability [&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],"tags":[],"class_list":["post-14409","post","type-post","status-publish","format-standard","hentry","category-all"],"modified_by":"IYATT-yx","_links":{"self":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/14409","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=14409"}],"version-history":[{"count":1,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/14409\/revisions"}],"predecessor-version":[{"id":19842,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/14409\/revisions\/19842"}],"wp:attachment":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14409"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14409"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14409"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}