{"id":304,"date":"2025-11-11T20:06:44","date_gmt":"2025-11-11T12:06:44","guid":{"rendered":"https:\/\/index.cmiteam.cn\/?p=304"},"modified":"2025-11-11T20:11:49","modified_gmt":"2025-11-11T12:11:49","slug":"%e5%b8%b8%e8%a7%81%e6%a6%82%e7%bb%9f%e5%88%86%e5%b8%83%e7%9a%84%e6%9c%9f%e6%9c%9b%ef%bc%8c%e6%96%b9%e5%b7%ae%e5%8f%8a%e5%85%ac%e5%bc%8f","status":"publish","type":"post","link":"https:\/\/index.cmiteam.cn\/index.php\/2025\/11\/11\/%e5%b8%b8%e8%a7%81%e6%a6%82%e7%bb%9f%e5%88%86%e5%b8%83%e7%9a%84%e6%9c%9f%e6%9c%9b%ef%bc%8c%e6%96%b9%e5%b7%ae%e5%8f%8a%e5%85%ac%e5%bc%8f\/","title":{"rendered":"\u5e38\u89c1\u6982\u7edf\u5206\u5e03\u7684\u671f\u671b\uff0c\u65b9\u5dee\u53ca\u516c\u5f0f"},"content":{"rendered":"\n<p><\/p>\n\n\n\n<p>\u597d\u7684\uff0c\u4e0b\u9762\u4e3a\u60a8\u603b\u7ed3\u4e86\u6b63\u6001\u5206\u5e03\u3001\u6307\u6570\u5206\u5e03\u3001\u5747\u5300\u5206\u5e03\u3001\u4f2f\u52aa\u5229\u5206\u5e03\u3001\u591a\u9879\u5206\u5e03\uff08\u591a\u91cd\u4f2f\u52aa\u5229\u5206\u5e03\u901a\u5e38\u6307\u591a\u9879\u5206\u5e03\uff09\u548c\u6cca\u677e\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\/\u8d28\u91cf\u51fd\u6570\u3001\u671f\u671b\u548c\u65b9\u5dee\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"486\" src=\"https:\/\/index.cmiteam.cn\/wp-content\/uploads\/2025\/11\/1762863105-Screenshot_2025-11-11-20-10-54-757_com.microsoft.emmx-edit-1024x486.jpg\" alt=\"\" class=\"wp-image-305\" srcset=\"https:\/\/index.cmiteam.cn\/wp-content\/uploads\/2025\/11\/1762863105-Screenshot_2025-11-11-20-10-54-757_com.microsoft.emmx-edit-1024x486.jpg 1024w, https:\/\/index.cmiteam.cn\/wp-content\/uploads\/2025\/11\/1762863105-Screenshot_2025-11-11-20-10-54-757_com.microsoft.emmx-edit-300x142.jpg 300w, https:\/\/index.cmiteam.cn\/wp-content\/uploads\/2025\/11\/1762863105-Screenshot_2025-11-11-20-10-54-757_com.microsoft.emmx-edit-768x364.jpg 768w, https:\/\/index.cmiteam.cn\/wp-content\/uploads\/2025\/11\/1762863105-Screenshot_2025-11-11-20-10-54-757_com.microsoft.emmx-edit-1536x729.jpg 1536w, https:\/\/index.cmiteam.cn\/wp-content\/uploads\/2025\/11\/1762863105-Screenshot_2025-11-11-20-10-54-757_com.microsoft.emmx-edit.jpg 1792w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">1. \u6b63\u6001\u5206\u5e03 (Normal Distribution)<\/h3>\n\n\n\n<p>\u6b63\u6001\u5206\u5e03\u662f\u8fde\u7eed\u578b\u6982\u7387\u5206\u5e03\uff0c\u7531\u4e24\u4e2a\u53c2\u6570\u51b3\u5b9a\uff1a\u5747\u503c (\\mu) \u548c\u65b9\u5dee (\\sigma^2)\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u6982\u7387\u5bc6\u5ea6\u51fd\u6570 (PDF)<\/strong>:<br>[ f(x | \\mu, \\sigma^2) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}} e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}} ]<br>\u5176\u4e2d\uff0c(x \\in (-\\infty, +\\infty))\u3002<\/li>\n\n\n\n<li><strong>\u671f\u671b (Expectation)<\/strong>:<br>[ E(X) = \\mu ]<\/li>\n\n\n\n<li><strong>\u65b9\u5dee (Variance)<\/strong>:<br>[ Var(X) = \\sigma^2 ]<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">2. \u6307\u6570\u5206\u5e03 (Exponential Distribution)<\/h3>\n\n\n\n<p>\u6307\u6570\u5206\u5e03\u662f\u8fde\u7eed\u578b\u6982\u7387\u5206\u5e03\uff0c\u5e38\u7528\u4e8e\u63cf\u8ff0\u72ec\u7acb\u968f\u673a\u4e8b\u4ef6\u53d1\u751f\u7684\u65f6\u95f4\u95f4\u9694\uff0c\u7531\u4e00\u4e2a\u53c2\u6570 (\\lambda) (\u7387\u53c2\u6570) \u51b3\u5b9a\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u6982\u7387\u5bc6\u5ea6\u51fd\u6570 (PDF)<\/strong>:<br>[ f(x | \\lambda) = \\lambda e^{-\\lambda x} ]<br>\u5176\u4e2d\uff0c(x \\ge 0) \u4e14 (\\lambda &gt; 0)\u3002<\/li>\n\n\n\n<li><strong>\u671f\u671b (Expectation)<\/strong>:<br>[ E(X) = \\frac{1}{\\lambda} ]<\/li>\n\n\n\n<li><strong>\u65b9\u5dee (Variance)<\/strong>:<br>[ Var(X) = \\frac{1}{\\lambda^2} ]<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">3. \u5747\u5300\u5206\u5e03 (Uniform Distribution)<\/h3>\n\n\n\n<p>\u5747\u5300\u5206\u5e03\u53ef\u4ee5\u662f\u8fde\u7eed\u578b\u6216\u79bb\u6563\u578b\u3002\u8fd9\u91cc\u6211\u4eec\u8ba8\u8bba\u8fde\u7eed\u578b\u5747\u5300\u5206\u5e03\uff0c\u5b83\u5728\u7ed9\u5b9a\u533a\u95f4 ([a, b]) \u5185\u7684\u4efb\u4f55\u4e00\u70b9\u7684\u6982\u7387\u5bc6\u5ea6\u90fd\u662f\u5e38\u6570\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u6982\u7387\u5bc6\u5ea6\u51fd\u6570 (PDF)<\/strong>:<br>[ f(x | a, b) = \\begin{cases} \\frac{1}{b-a} &amp; \\text{for } a \\le x \\le b \\ 0 &amp; \\text{otherwise} \\end{cases} ]<br>\u5176\u4e2d\uff0c(a &lt; b)\u3002<\/li>\n\n\n\n<li><strong>\u671f\u671b (Expectation)<\/strong>:<br>[ E(X) = \\frac{a+b}{2} ]<\/li>\n\n\n\n<li><strong>\u65b9\u5dee (Variance)<\/strong>:<br>[ Var(X) = \\frac{(b-a)^2}{12} ]<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">4. \u4f2f\u52aa\u5229\u5206\u5e03 (Bernoulli Distribution)<\/h3>\n\n\n\n<p>\u4f2f\u52aa\u5229\u5206\u5e03\u662f\u79bb\u6563\u578b\u6982\u7387\u5206\u5e03\uff0c\u63cf\u8ff0\u5355\u6b21\u8bd5\u9a8c\u53ea\u6709\u4e24\u79cd\u53ef\u80fd\u7ed3\u679c\uff08\u6210\u529f\u6216\u5931\u8d25\uff09\u7684\u60c5\u51b5\uff0c\u7531\u53c2\u6570 (p) (\u6210\u529f\u6982\u7387) \u51b3\u5b9a\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u6982\u7387\u8d28\u91cf\u51fd\u6570 (PMF)<\/strong>:<br>[ P(X=k | p) = \\begin{cases} p &amp; \\text{if } k=1 \\ 1-p &amp; \\text{if } k=0 \\end{cases} ]<br>\u4e5f\u53ef\u4ee5\u5199\u6210\uff1a<br>[ P(X=k | p) = p^k (1-p)^{1-k} ]<br>\u5176\u4e2d\uff0c(k \\in {0, 1}) \u4e14 (0 \\le p \\le 1)\u3002<\/li>\n\n\n\n<li><strong>\u671f\u671b (Expectation)<\/strong>:<br>[ E(X) = p ]<\/li>\n\n\n\n<li><strong>\u65b9\u5dee (Variance)<\/strong>:<br>[ Var(X) = p(1-p) ]<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">5. \u591a\u9879\u5206\u5e03 (Multinomial Distribution)<\/h3>\n\n\n\n<p>\u591a\u9879\u5206\u5e03\u662f\u79bb\u6563\u578b\u6982\u7387\u5206\u5e03\uff0c\u662f\u4f2f\u52aa\u5229\u5206\u5e03\u7684\u63a8\u5e7f\uff0c\u63cf\u8ff0\u8fdb\u884c (n) \u6b21\u72ec\u7acb\u8bd5\u9a8c\uff0c\u6bcf\u6b21\u8bd5\u9a8c\u6709 (k) \u79cd\u53ef\u80fd\u7ed3\u679c\uff0c\u6bcf\u79cd\u7ed3\u679c\u6709\u56fa\u5b9a\u7684\u6982\u7387 (p_1, \\dots, p_k)\u3002\u5b83\u7ed9\u51fa\u4e86\u6bcf\u79cd\u7ed3\u679c\u51fa\u73b0\u6b21\u6570\u7684\u8054\u5408\u6982\u7387\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u6982\u7387\u8d28\u91cf\u51fd\u6570 (PMF)<\/strong>:<br>[ P(X_1=x_1, \\dots, X_k=x_k | n, p_1, \\dots, p_k) = \\frac{n!}{x_1!x_2!\\dots x_k!} p_1^{x_1} p_2^{x_2} \\dots p_k^{x_k} ]<br>\u5176\u4e2d\uff0c(\\sum_{i=1}^k x_i = n)\uff0c(x_i \\ge 0) \u5747\u4e3a\u6574\u6570\uff0c(\\sum_{i=1}^k p_i = 1)\uff0c\u4e14 (p_i \\ge 0)\u3002<\/li>\n\n\n\n<li><strong>\u671f\u671b (Expectation)<\/strong> (\u5bf9\u4e8e\u6bcf\u4e2a\u7c7b\u522b (i) \u7684\u51fa\u73b0\u6b21\u6570 (X_i)):<br>[ E(X_i) = n p_i ]<\/li>\n\n\n\n<li><strong>\u65b9\u5dee (Variance)<\/strong> (\u5bf9\u4e8e\u6bcf\u4e2a\u7c7b\u522b (i) \u7684\u51fa\u73b0\u6b21\u6570 (X_i)):<br>[ Var(X_i) = n p_i (1-p_i) ]<br><em>\u6ce8\u610f\uff1a\u591a\u9879\u5206\u5e03\u8fd8\u6d89\u53ca\u534f\u65b9\u5dee\uff0c\u5bf9\u4e8e (i \\ne j)\uff0c(Cov(X_i, X_j) = -n p_i p_j)\u3002<\/em><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">6. \u6cca\u677e\u5206\u5e03 (Poisson Distribution)<\/h3>\n\n\n\n<p>\u6cca\u677e\u5206\u5e03\u662f\u79bb\u6563\u578b\u6982\u7387\u5206\u5e03\uff0c\u63cf\u8ff0\u5728\u56fa\u5b9a\u65f6\u95f4\u6216\u7a7a\u95f4\u95f4\u9694\u5185\uff0c\u4e8b\u4ef6\u53d1\u751f\u6b21\u6570\u7684\u6982\u7387\uff0c\u7531\u53c2\u6570 (\\lambda) (\u5e73\u5747\u53d1\u751f\u7387) \u51b3\u5b9a\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u6982\u7387\u8d28\u91cf\u51fd\u6570 (PMF)<\/strong>:<br>[ P(X=k | \\lambda) = \\frac{e^{-\\lambda} \\lambda^k}{k!} ]<br>\u5176\u4e2d\uff0c(k \\in {0, 1, 2, \\dots}) (\u975e\u8d1f\u6574\u6570) \u4e14 (\\lambda &gt; 0)\u3002<\/li>\n\n\n\n<li><strong>\u671f\u671b (Expectation)<\/strong>:<br>[ E(X) = \\lambda ]<\/li>\n\n\n\n<li><strong>\u65b9\u5dee (Variance)<\/strong>:<br>[ Var(X) = \\lambda ]<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n","protected":false},"excerpt":{"rendered":"<p>\u597d\u7684\uff0c\u4e0b\u9762\u4e3a\u60a8\u603b\u7ed3\u4e86\u6b63\u6001\u5206\u5e03\u3001\u6307\u6570\u5206\u5e03\u3001\u5747\u5300\u5206\u5e03\u3001\u4f2f\u52aa\u5229\u5206\u5e03\u3001\u591a\u9879\u5206\u5e03\uff08\u591a\u91cd\u4f2f\u52aa\u5229\u5206\u5e03\u901a\u5e38\u6307\u591a\u9879\u5206\u5e03\uff09\u548c\u6cca\u677e\u5206\u5e03\u7684\u6982\u7387\u5bc6\u5ea6\/\u8d28\u91cf\u51fd ...<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"emotion":"","emotion_color":"","title_style":"","license":""},"categories":[1],"tags":[],"class_list":["post-304","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/index.cmiteam.cn\/index.php\/wp-json\/wp\/v2\/posts\/304","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/index.cmiteam.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/index.cmiteam.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/index.cmiteam.cn\/index.php\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/index.cmiteam.cn\/index.php\/wp-json\/wp\/v2\/comments?post=304"}],"version-history":[{"count":0,"href":"https:\/\/index.cmiteam.cn\/index.php\/wp-json\/wp\/v2\/posts\/304\/revisions"}],"wp:attachment":[{"href":"https:\/\/index.cmiteam.cn\/index.php\/wp-json\/wp\/v2\/media?parent=304"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/index.cmiteam.cn\/index.php\/wp-json\/wp\/v2\/categories?post=304"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/index.cmiteam.cn\/index.php\/wp-json\/wp\/v2\/tags?post=304"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}