{"id":2520,"date":"2026-09-18T08:36:40","date_gmt":"2026-09-18T08:36:40","guid":{"rendered":"https:\/\/dev95.site\/%d9%85%d8%b3%d8%a3%d9%84%d8%a9-%d8%ac%d8%a7%d9%85%d8%b9-%d8%a7%d9%84%d9%82%d8%b3%d8%a7%d8%a6%d9%85\/"},"modified":"2026-09-18T08:36:40","modified_gmt":"2026-09-18T08:36:40","slug":"%d9%85%d8%b3%d8%a3%d9%84%d8%a9-%d8%ac%d8%a7%d9%85%d8%b9-%d8%a7%d9%84%d9%82%d8%b3%d8%a7%d8%a6%d9%85","status":"publish","type":"post","link":"https:\/\/dev95.site\/ar\/%d9%85%d8%b3%d8%a3%d9%84%d8%a9-%d8%ac%d8%a7%d9%85%d8%b9-%d8%a7%d9%84%d9%82%d8%b3%d8%a7%d8%a6%d9%85\/","title":{"rendered":"\u0645\u0633\u0623\u0644\u0629 \u062c\u0627\u0645\u0639 \u0627\u0644\u0642\u0633\u0627\u0626\u0645"},"content":{"rendered":"<div id=\"dev95-1959951602\" class=\"dev95-- dev95-entity-placement\"><script async=\"async\" data-cfasync=\"false\" src=\"https:\/\/pl27862732.profitableratecpmnetwork.com\/2ad7a50e0bbc23ac6801d7b77c501463\/invoke.js\"><\/script>\r\n<div id=\"container-2ad7a50e0bbc23ac6801d7b77c501463\"><\/div><\/div><div>\n<p>SanBonne: <\/p>\n<hr>\n<div>{{\u0635\u0646\u062f\u0648\u0642 \u0645\u0639\u0644\u0648\u0645\u0627\u062a \u062a\u0648\u0632\u064a\u0639 \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u0627\u062a<br \/>\n  | name       = \u0645\u0633\u0623\u0644\u0629 \u062c\u0627\u0645\u0639 \u0627\u0644\u0642\u0633\u0627\u0626\u0645<br \/>\n  | type       = \u0645\u0646\u0641\u0635\u0644<br \/>\n  | parameters = &lt;math&gt;ninmathbb N&lt;\/math&gt; &amp;ndash; \u0639\u062f\u062f \u0623\u0648\u062c\u0647 \u0627\u0644\u0646\u0631\u062f<br \/>\n  | support    = &lt;math&gt;kinmathbb N&lt;\/math&gt; &amp;ndash; \u0627\u0644\u0623\u062f\u0648\u0627\u0631 \u0627\u0644\u0644\u0627\u0632\u0645\u0629 \u0644\u062a\u0638\u0647\u0631 \u0627\u0644\u0623\u0648\u062c\u0647 \u062c\u0645\u064a\u0639\u0647\u0627<br \/>\n  &lt;!&#8211; | pdf        = &lt;math&gt;frac{(n-1)^{{k-1}}}{n^{k-1}}&lt;\/math&gt; &#8211;&gt;<br \/>\n  | cdf        = &lt;math&gt;frac{n^{{k}}}{n^k}&lt;\/math&gt;<br \/>\n  | mean       = &lt;math&gt;nH_n&lt;\/math&gt;<br \/>\n  &lt;!&#8211; | variance   = &lt;math&gt;n^2H^{(2)}_n-nH_n&lt;\/math&gt; &#8211;&gt;<br \/>\n | \u0627\u0644\u062a\u0641\u0631\u0637\u062d   = &lt;math&gt;frac{2n^3H^{(3)}_n-3n^2H^{(2)}_n+nH_n}{left(n^2H^{(2)}_n-nH_nright)^{3\/2}} underset nsim 6^{3\/2}2frac{zeta(3)}{pi^3}&lt;\/math&gt;<br \/>\n  | kurtosis   = &lt;math&gt;frac{6n^4H^{(4)}_n-12n^3H^{(3)}_n+7n^2H^{(2)}_n-nH_n}{(n^2H^{(2)}_n-nH_n)^2}simfrac65&lt;\/math&gt;<br \/>\n  | pgf        = &lt;math&gt;G(z) = {frac nzchoose n}^{-1}&lt;\/math&gt;<br \/>\n  | mgf        = &lt;math&gt;{frac n{e^t}choose n}^{-1}&lt;\/math&gt;<br \/>\n  | char       = &lt;math&gt;{frac n{e^{it}}choose n}^{-1}&lt;\/math&gt;<br \/>\n}}<br \/>\n[[File:Coupon collector problem.svg|thumb|400px|\u0631\u0633\u0645 \u0628\u064a\u0627\u0646\u064a \u0644\u0639\u062f\u062f \u0627\u0644\u0642\u0633\u0627\u0626\u0645  {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a \u0645\u0627\u0626\u0644|n}} \u0645\u0642\u0627\u0628\u0644 \u0639\u062f\u062f \u0627\u0644\u062f\u0648\u0631\u0627\u062a (\u0645\u062b\u0644 \u0627\u0644\u0632\u0645\u0646) \u0627\u0644\u0644\u0627\u0632\u0645\u0629 \u0644\u062c\u0645\u0639\u0647\u0645 \u062c\u0645\u064a\u0639\u064b\u0627 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a|&#8221;E&#8221;(&#8221;T&#8221;)}}]]<br \/>\n\u062a\u0634\u064a\u0631 &#8221;&#8217;\u0645\u0633\u0623\u0644\u0629 \u062c\u0627\u0645\u0639 \u0627\u0644\u0642\u0633\u0627\u0626\u0645&#8221;&#8217; {{\u0625\u0646\u062c|Coupon collector&#8217;s problem}} \u0641\u064a [[\u0646\u0638\u0631\u064a\u0629 \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644]] \u0625\u0644\u0649 \u0627\u0644\u062a\u062d\u0644\u064a\u0644 \u0627\u0644\u0631\u064a\u0627\u0636\u064a \u0644\u0645\u0633\u0627\u0628\u0642\u0627\u062a &#8220;\u0627\u062c\u0645\u0639 \u062c\u0645\u064a\u0639 [[\u0642\u0633\u064a\u0645\u0629|\u0627\u0644\u0642\u0633\u0627\u0626\u0645]] \u0648\u0641\u0632&#8221;\u060c \u0625\u0630 \u062a\u0637\u0631\u062d \u0627\u0644\u0645\u0633\u0623\u0644\u0629 \u0627\u0644\u0633\u0624\u0627\u0644 \u0627\u0644\u0622\u062a\u064a: \u0625\u0630\u0627 \u0627\u062d\u062a\u0648\u062a \u0643\u0644 \u0639\u0644\u0628\u0629 \u0645\u0646\u062a\u062c \u0645\u0639\u064a\u0646 (\u0645\u062b\u0644 \u062d\u0628\u0648\u0628 \u0627\u0644\u0625\u0641\u0637\u0627\u0631) \u0642\u0633\u064a\u0645\u0629\u064b\u060c \u0648\u062a\u0648\u0641\u0631\u062a \u0623\u0646\u0648\u0627\u0639 \u0645\u062a\u0639\u062f\u062f\u0629 \u0645\u0646 \u0627\u0644\u0642\u0633\u0627\u0626\u0645\u060c \u0641\u0645\u0627 \u0627\u062d\u062a\u0645\u0627\u0644 \u0634\u0631\u0627\u0621 \u0623\u0643\u062b\u0631 \u0645\u0646 \u0639\u062f\u062f \u0645\u0639\u064a\u0646 \u0645\u0646 \u0627\u0644\u0639\u0644\u0628 \u0644\u062c\u0645\u0639 \u0627\u0644\u0642\u0633\u0627\u0626\u0645 \u0627\u0644\u0645\u062a\u0646\u0648\u0639\u0629 \u0643\u064f\u0644\u064e\u0651\u0647\u0627\u061f<\/p>\n<p>\u0623\u064a\u0636\u064b\u0627\u060c \u064a\u0645\u0643\u0646 \u0635\u064a\u0627\u063a\u0629 \u0647\u0630\u0647 \u0627\u0644\u0645\u0633\u0623\u0644\u0629 \u0635\u064a\u0627\u063a\u0629 \u0623\u062e\u0631\u0649 \u0648\u0647\u064a: \u0625\u0630\u0627 \u062a\u0648\u0641\u0631 \u0639\u062f\u062f \u0645\u062d\u062f\u062f (\u0641\u0631\u0636\u064b\u0627 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a \u0645\u0627\u0626\u0644|n}}) \u0645\u0646 \u0623\u0646\u0648\u0627\u0639 \u0627\u0644\u0642\u0633\u0627\u0626\u0645\u060c  \u0641\u0645\u0627 [[\u0642\u064a\u0645\u0629 \u0645\u062a\u0648\u0642\u0639\u0629|\u0627\u0644\u0642\u064a\u0645\u0629 \u0627\u0644\u0645\u062a\u0648\u0642\u0639\u0629]] \u0644\u0644\u0642\u0633\u0627\u0626\u0645 \u0627\u0644\u0648\u0627\u062c\u0628 \u0633\u062d\u0628\u0647\u0627 \u0645\u0639 [[\u0627\u0639\u062a\u064a\u0627\u0646|\u0625\u0639\u0627\u062f\u0629 \u0627\u0644\u0642\u0633\u0627\u0626\u0645 \u0625\u0644\u0649 \u0627\u0644\u0645\u062c\u0645\u0648\u0639\u0629]] \u0642\u0628\u0644 \u0623\u0646 \u062a\u0633\u062d\u0628 \u0643\u0644 \u0642\u0633\u064a\u0645\u0629 \u0645\u0631\u0629 \u0648\u0627\u062d\u062f\u0629 \u0639\u0644\u0649 \u0627\u0644\u0623\u0642\u0644\u061f<\/p><div id=\"dev95-3896040396\" class=\"dev95- dev95-entity-placement\"><center>\r\n<script>\r\n  atOptions = {\r\n    'key' : '4ba6b6513c00e0ba76511f798ae56401',\r\n    'format' : 'iframe',\r\n    'height' : 50,\r\n    'width' : 320,\r\n    'params' : {}\r\n  };\r\n<\/script>\r\n<script src=\"https:\/\/www.highrevenueformat.com\/4ba6b6513c00e0ba76511f798ae56401\/invoke.js\"><\/script>\r\n\t<\/center><\/div>\n<p>\u064a\u064f\u0638\u0647\u0631 \u0627\u0644\u062a\u062d\u0644\u064a\u0644 \u0627\u0644\u0631\u064a\u0627\u0636\u064a \u0644\u0644\u0645\u0633\u0623\u0644\u0629 \u0623\u0646 [[\u0642\u064a\u0645\u0629 \u0645\u062a\u0648\u0642\u0639\u0629|\u0627\u0644\u0642\u064a\u0645\u0629 \u0627\u0644\u0645\u062a\u0648\u0642\u0639\u0629]] \u0644\u0639\u062f\u062f \u0627\u0644\u0645\u062d\u0627\u0648\u0644\u0627\u062a \u0627\u0644\u0645\u0637\u0644\u0648\u0628\u0629 \u062a\u0632\u062f\u0627\u062f \u0628\u0645\u0639\u062f\u0644 &lt;math&gt;Theta(nlog(n))&lt;\/math&gt;. {{\u0645\u0644\u0627|\u062a\u0634\u064a\u0631 [[\u0644\u0648\u063a\u0627\u0631\u064a\u062a\u0645|\u0627\u0644\u0644\u0648\u063a\u0627\u0631\u062a\u0645]] \u0647\u0646\u0627 \u0648\u0641\u064a \u0627\u0644\u0645\u0642\u0627\u0644\u0629 \u0643\u0643\u0644 \u0625\u0644\u0649 [[\u0644\u063a\u0627\u0631\u062a\u0645 \u0637\u0628\u064a\u0639\u064a|\u0627\u0644\u0644\u0648\u063a\u0627\u0631\u062a\u0645 \u0627\u0644\u0637\u0628\u064a\u0639\u064a]] \u0644\u0627 \u0625\u0644\u0649 \u0644\u0648\u063a\u0627\u0631\u062a\u0645 \u0644\u0623\u0633\u0627\u0633 \u0622\u062e\u0631. \u0648\u064a\u0633\u062a\u062f\u0639\u064a \u0627\u0633\u062a\u0639\u0645\u0627\u0644 \u0627\u0644\u0631\u0645\u0632 \u0398 \u0647\u0646\u0627 [[\u062a\u0645\u062b\u064a\u0644 O \u0627\u0644\u0643\u0628\u0631\u0649]].}} \u062a\u0633\u062a\u0644\u0632\u0645 \u0627\u0644\u0639\u0645\u0644\u064a\u0629 \u0645\u062b\u0644\u0627\u064b \u0639\u0646\u062f \u062a\u0648\u0641\u0631 50 \u0646\u0648\u0639\u064b\u0627 \u0645\u062e\u062a\u0644\u0641\u064b\u0627 \u0646\u062d\u0648 225 \u0645\u062d\u0627\u0648\u0644\u0629\u060c{{\u0645\u0644\u0627|1= &lt;math&gt;E(50) = 50(1 + 1\/2 + 1\/3 + &#8230; + 1\/50) = 224.9603&lt;\/math&gt; \u0647\u0648 \u0627\u0644\u0639\u062f\u062f \u0627\u0644\u0645\u062a\u0648\u0642\u0639 \u0645\u0646 \u0627\u0644\u0645\u062d\u0627\u0648\u0644\u0627\u062a \u0627\u0644\u0644\u0627\u0632\u0645\u0629 \u0644\u062c\u0645\u0639 \u062c\u0645\u064a\u0639 \u0627\u0644\u0642\u0633\u0627\u0626\u0645 \u0627\u0644\u062e\u0645\u0633\u064a\u0646. \u0627\u0644\u062a\u0642\u062f\u064a\u0631 \u0627\u0644\u062a\u0642\u0631\u064a\u0628\u064a: &lt;math&gt;nlog n+gamma n+1\/2&lt;\/math&gt;&lt;br&gt; \u0648\u064a\u064f\u0639\u0637\u064a \u0627\u0644\u062a\u0642\u062f\u064a\u0631 \u0627\u0644\u062a\u0642\u0631\u064a\u0628\u064a \u0644\u0647\u0630\u0627 \u0627\u0644\u0639\u062f\u062f \u0627\u0644\u0645\u062a\u0648\u0642\u0639 \u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u062d\u0627\u0644\u0629:&lt;br&gt;&lt;math&gt;50log 50+50gamma+1\/2 approx 195.6011+28.8608+0.5approx 224.9619&lt;\/math&gt;.}} \u0644\u062c\u0645\u0639 \u062c\u0645\u064a\u0639 \u0627\u0644\u0642\u0633\u0627\u0626\u0645 \u0627\u0644\u062e\u0645\u0633\u064a\u0646. \u0648\u0623\u062d\u064a\u0627\u0646\u064b\u0627 \u062a\u064f\u0639\u0628\u0631 \u0639\u0646 \u0627\u0644\u0645\u0634\u0643\u0644\u0629 \u0628\u062f\u0644\u0627\u064b \u0645\u0646 \u0630\u0644\u0643 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0646\u0631\u062f \u0645\u062a\u0639\u062f\u062f \u0627\u0644\u0623\u0648\u062c\u0647 (\u0630\u0648 n \u0648\u062c\u0647\u064b\u0627).<br \/>\n== \u0648\u0635\u0641 \u0627\u0644\u0645\u0633\u0623\u0644\u0629 ==<br \/>\n\u064a\u0647\u062f\u0641 \u0627\u0644\u062c\u0627\u0645\u0639 \u0644\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 \u0642\u0633\u064a\u0645\u0629 \u0648\u0627\u062d\u062f\u0629 \u0645\u0646 \u0643\u0644 \u0646\u0648\u0639 \u0645\u0646 \u0623\u0646\u0648\u0627\u0639 \u0627\u0644\u0642\u0633\u0627\u0626\u0645 \u0648\u0639\u062f\u062f\u0647\u0627 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a \u0645\u0627\u0626\u0644|n}}. \u0644\u062a\u062d\u0642\u064a\u0642 \u0630\u0644\u0643\u060c \u064a\u0634\u062a\u0631\u064a \u0639\u0644\u0628\u064b\u0627 \u0645\u0646 \u062d\u0628\u0648\u0628 \u0627\u0644\u0625\u0641\u0637\u0627\u0631. \u0648\u0643\u0645\u0627 \u064a\u0648\u0636\u062d \u0627\u0644\u0634\u0643\u0644 \u0623\u062f\u0646\u0627\u0647\u060c \u062a\u062d\u062a\u0648\u064a \u0643\u0644 \u0639\u0644\u0628\u0629 \u0639\u0644\u0649 \u0642\u0633\u064a\u0645\u0629 \u0648\u0627\u062d\u062f\u0629. \u0641\u064a \u0627\u0644\u0645\u062b\u0627\u0644\u060c \u064a\u0634\u062a\u0631\u064a \u0627\u0644\u0639\u0644\u0628\u0629 \u0627\u0644\u0623\u0648\u0644\u0649 \u0648\u064a\u062d\u0635\u0644 \u0641\u064a\u0647\u0627 \u0639\u0644\u0649 \u0627\u0644\u0642\u0633\u064a\u0645\u0629 1. \u062b\u0645 \u064a\u0634\u062a\u0631\u064a \u0627\u0644\u0639\u0644\u0628\u0629 \u0627\u0644\u062b\u0627\u0646\u064a\u0629 \u0648\u064a\u062d\u0635\u0644 \u0639\u0644\u0649 \u0627\u0644\u0642\u0633\u064a\u0645\u0629 4. \u0648\u0641\u064a \u0639\u0645\u0644\u064a\u0629 \u0627\u0644\u0634\u0631\u0627\u0621 \u0627\u0644\u062b\u0627\u0644\u062b\u0629\u060c \u064a\u062d\u0635\u0644 \u0639\u0644\u0649 \u0627\u0644\u0642\u0633\u064a\u0645\u0629 1 \u0645\u0631\u0629 \u0623\u062e\u0631\u0649\u060c \u0648\u0644\u0630\u0644\u0643 \u0644\u0627 \u062a\u062a\u063a\u064a\u0631 \u0645\u062c\u0645\u0648\u0639\u062a\u0647\u061b \u0625\u0630 \u064a\u0638\u0644 \u064a\u0645\u062a\u0644\u0643 \u0627\u0644\u0642\u0633\u064a\u0645\u0629\u064a\u0646 1 \u06484. \u0648\u064a\u0648\u0627\u0635\u0644 \u0634\u0631\u0627\u0621 \u0639\u0644\u0628 \u062d\u0628\u0648\u0628 \u0627\u0644\u0625\u0641\u0637\u0627\u0631 \u062d\u062a\u0649 \u064a\u062d\u0635\u0644 \u0639\u0644\u0649 \u0645\u0644\u0635\u0642 \u0645\u0646 \u0643\u0644 \u0646\u0648\u0639. \u0648\u0646\u0641\u062a\u0631\u0636 \u0623\u0646 \u062c\u0645\u064a\u0639 \u0627\u0644\u0642\u0633\u0627\u0626\u0645 \u0645\u062a\u0633\u0627\u0648\u064a\u0629 \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u060c \u0623\u064a \u0625\u0646 \u0627\u062d\u062a\u0645\u0627\u0644 \u0627\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 \u0623\u064a \u0646\u0648\u0639 \u0645\u0646 \u0623\u0646\u0648\u0627\u0639 \u0627\u0644\u0642\u0633\u0627\u0626\u0645 \u0639\u0646\u062f \u0634\u0631\u0627\u0621 \u0639\u0644\u0628\u0629 \u0647\u0648 {{\u0643\u0633\u0631|1|&#8221;n&#8221;}}.<\/p>\n<p>\u0646\u0631\u0645\u0632 \u0625\u0644\u0649 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a \u0645\u0627\u0626\u0644|T&lt;sub&gt;n&lt;\/sub&gt;}} \u0628\u0639\u062f\u062f \u0627\u0644\u0639\u0644\u0628 \u0627\u0644\u0644\u0627\u0632\u0645 \u0634\u0631\u0627\u0624\u0647\u0627 \u0644\u062c\u0645\u0639 \u0627\u0644\u0642\u0633\u0627\u0626\u0645 \u0643\u0627\u0645\u0644\u0629\u064b. \u0641\u0641\u064a \u0627\u0644\u0645\u062b\u0627\u0644 \u0623\u0639\u0644\u0627\u0647\u060c &lt;math&gt;T_4=10&lt;\/math&gt;\u060c \u0644\u0623\u0646 \u0627\u0644\u062c\u0627\u0645\u0639 \u0627\u0634\u062a\u0631\u0649 10 \u0639\u0644\u0628 \u0645\u0646 \u062d\u0628\u0648\u0628 \u0627\u0644\u0625\u0641\u0637\u0627\u0631 \u0642\u0628\u0644 \u0623\u0646 \u064a\u062d\u0635\u0644 \u0639\u0644\u0649 \u0627\u0644\u0642\u0633\u0627\u0626\u0645 \u0627\u0644\u0623\u0631\u0628\u0639. \u0648\u0645\u0646 \u0627\u0644\u0628\u062f\u064a\u0647\u064a \u0623\u0646 \u0627\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 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\u0632\u0645\u0646\u064a\u0629.<br \/>\n[[\u0645\u0644\u0641:Coupon collectors problem example.svg|\u0645\u0631\u0643\u0632|\u0625\u0637\u0627\u0631|1000 \u0628\u0643|\u0645\u062b\u0627\u0644 \u0639\u0644\u0649 \u0645\u062c\u0645\u0648\u0639\u0629 \u0645\u0646 4 \u0642\u0633\u0627\u0626\u0645 \u0645\u0645\u0643\u0646\u0629: 1\u060c 2\u060c 3\u060c 4. \u0641\u064a \u0627\u0644\u0645\u062b\u0627\u0644\u060c \u0646\u062d\u0635\u0644 \u0641\u064a \u0627\u0644\u062e\u0637\u0648\u0629 1 \u0639\u0644\u0649 \u0627\u0644\u0642\u0633\u064a\u0645\u0629 1\u060c \u0648\u0641\u064a \u0627\u0644\u062e\u0637\u0648\u0629 2 \u0639\u0644\u0649 \u0627\u0644\u0642\u0633\u064a\u0645\u0629 4\u060c \u0648\u0641\u064a \u0627\u0644\u062e\u0637\u0648\u0629 3 \u0639\u0644\u0649 \u0627\u0644\u0642\u0633\u064a\u0645\u0629 1 \u0645\u0631\u0629 \u0623\u062e\u0631\u0649\u060c \u0648\u0647\u0643\u0630\u0627.]]<br \/>\n\u0644\u062a\u0643\u0646 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a \u0645\u0627\u0626\u0644|t&lt;sub&gt;n,i&lt;\/sub&gt;}} \u0627\u0644\u0632\u0645\u0646 \u0627\u0644\u0625\u0636\u0627\u0641\u064a \u0627\u0644\u0644\u0627\u0632\u0645 \u0644\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 \u0627\u0644\u0642\u0633\u064a\u0645\u0629 \u0627\u0644\u062c\u062f\u064a\u062f\u0629 \u0631\u0642\u0645 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a \u0645\u0627\u0626\u0644|i}}\u060c \u0645\u0639 \u0627\u0644\u0639\u0644\u0645 \u0623\u0646\u0646\u0627 \u0646\u0645\u062a\u0644\u0643 \u0628\u0627\u0644\u0641\u0639\u0644 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a|&#8221;i&#8221; \u2013 1}} \u0642\u0633\u064a\u0645\u0629 \u0645\u062e\u062a\u0644\u0641\u0629. \u0648\u0645\u0646 \u062b\u0645 &lt;math&gt;T_{n,k}=sum_{i=1}^k t_{n,i}&lt;\/math&gt;. \u0648\u0641\u064a \u0645\u062b\u0627\u0644 \u0627\u0644\u0634\u0643\u0644 \u0623\u0639\u0644\u0627\u0647\u060c \u0644\u062f\u064a\u0646\u0627 &lt;math&gt;t_{4,1} = 1&lt;\/math&gt; (\u0648\u064a\u0643\u0648\u0646 &lt;math&gt;t_{n,1}&lt;\/math&gt; \u0645\u0633\u0627\u0648\u064a\u064b\u0627 \u062f\u0627\u0626\u0645\u064b\u0627 \u0644\u06401)\u060c \u0648&lt;math&gt;t_{4,2} = 1&lt;\/math&gt;\u060c \u0648&lt;math&gt;t_{4,3} = 3&lt;\/math&gt;\u060c \u0648&lt;math&gt;t_{4,4} = 5&lt;\/math&gt;. \u0648\u0647\u0646\u0627 \u0644\u062f\u064a\u0646\u0627 &lt;math&gt;T_{4}=10&lt;\/math&gt; \u0625\u0630 \u0627\u0634\u062a\u0631\u0649 \u0627\u0644\u062c\u0627\u0645\u0639 \u0641\u064a \u0627\u0644\u0645\u062b\u0627\u0644 10 \u0639\u0644\u0628 \u0644\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 \u0627\u0644\u0645\u062c\u0645\u0648\u0639\u0629 \u0627\u0644\u0643\u0627\u0645\u0644\u0629 \u0627\u0644\u0645\u0643\u0648\u0651\u0646\u0629 \u0645\u0646 \u0627\u0644\u0642\u0633\u0627\u0626\u0645 \u0627\u0644\u0623\u0631\u0628\u0639.<br \/>\n==\u0627\u0644\u062d\u0644==<br \/>\n===\u062d\u0633\u0627\u0628 \u0627\u0644\u0642\u064a\u0645\u0629 \u0627\u0644\u0645\u062a\u0648\u0642\u0639\u0629===<br \/>\n\u0644\u064a\u0643\u0646 \u0627\u0644\u0632\u0645\u0646 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a \u0645\u0627\u0626\u0644|T}} \u0647\u0648 \u0627\u0644\u0632\u0645\u0646 \u0627\u0644\u0644\u0627\u0632\u0645 \u0644\u062c\u0645\u0639 \u062c\u0645\u064a\u0639 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a \u0645\u0627\u0626\u0644|n}} \u0642\u0633\u0627\u0626\u0645\u060c \u0648\u0644\u064a\u0643\u0646 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a|&#8221;t&#8221;&lt;sub&gt;&#8221;i&#8221;&lt;\/sub&gt;}} \u0627\u0644\u0632\u0645\u0646\u064e \u0627\u0644\u0644\u0627\u0632\u0645 \u0644\u062c\u0645\u0639 \u0627\u0644\u0642\u0633\u064a\u0645\u0629 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a \u0645\u0627\u0626\u0644|i}} \u0628\u0639\u062f \u062c\u0645\u0639 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a|&#8221;i&#8221; \u2212 1}} \u0642\u0633\u064a\u0645\u0629. \u0639\u0646\u062f\u0626\u0630\u064d &lt;math&gt;T=t_1 + cdots + t_n&lt;\/math&gt;. \u064a\u064f\u0639\u062f\u0651 \u0643\u0644\u064c\u0651 \u0645\u0646 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a \u0645\u0627\u0626\u0644|T}} \u0648{{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a|&#8221;t&#8221;&lt;sub&gt;&#8221;i&#8221;&lt;\/sub&gt;}} [[\u0645\u062a\u063a\u064a\u0631 \u0639\u0634\u0648\u0627\u0626\u064a|\u0645\u062a\u063a\u064a\u0631\u064a\u0646 \u0639\u0634\u0648\u0627\u0626\u064a\u064a\u0646]]. \u0648\u064a\u064f\u0644\u0627\u062d\u064e\u0638 \u0623\u0646 \u0627\u062d\u062a\u0645\u0627\u0644 \u0627\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 \u0627\u0644\u0642\u0633\u064a\u0645\u0629 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a \u0645\u0627\u0626\u0644|i}} &#8221;&#8217;\u0627\u0644\u062c\u062f\u064a\u062f\u0629&#8221;&#8217; \u0647\u0648 &lt;math&gt;p_i = frac{n &#8211; (i &#8211; 1)}{n} = frac{n &#8211; i + 1}n&lt;\/math&gt;. \u0648\u0645\u0646 \u062b\u064e\u0645\u064e\u0651 \u064a\u062a\u0628\u0639 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a|&#8221;t&#8221;&lt;sub&gt;&#8221;i&#8221;&lt;\/sub&gt;}} [[\u062a\u0648\u0632\u064a\u0639 \u0647\u0646\u062f\u0633\u064a|\u0627\u0644\u062a\u0648\u0632\u064a\u0639\u064e \u0627\u0644\u0647\u0646\u062f\u0633\u064a]] \u0628\u0642\u064a\u0645\u0629 \u0645\u062a\u0648\u0642\u0639\u0629 &lt;math&gt;frac{1}{p_i} = frac n{n &#8211; i + 1}&lt;\/math&gt;. \u0648\u0628\u062a\u0637\u0628\u064a\u0642 [[\u0642\u064a\u0645\u0629 \u0645\u062a\u0648\u0642\u0639\u0629|\u062e\u0637\u064a\u0629 \u0627\u0644\u0642\u064a\u0645\u0629 \u0627\u0644\u0645\u062a\u0648\u0642\u0639\u0629]] \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/p>\n<p>: &lt;math&gt;<br \/>\nbegin{align}<br \/>\noperatorname{E}(T) &amp; {}= operatorname{E}(t_1 + t_2 + cdots + t_n) \\<br \/>\n&amp; {}= operatorname{E}(t_1) + operatorname{E}(t_2) + cdots + operatorname{E}(t_n) \\<br \/>\n&amp; {}= frac{1}{p_1} + frac{1}{p_2} +  cdots + frac{1}{p_n} \\<br \/>\n&amp; {}= frac{n}{n} + frac{n}{n-1} +  cdots + frac{n}{1} \\<br \/>\n&amp; {}= n cdot left(frac{1}{1} + frac{1}{2} + cdots + frac{1}{n}right) \\<br \/>\n&amp; {}= n cdot H_n.<br \/>\nend{align}<br \/>\n&lt;\/math&gt;<\/p>\n<p>\u062d\u064a\u062b {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a|&#8221;H&#8221;&lt;sub&gt;&#8221;n&#8221;&lt;\/sub&gt;}} \u0647\u0648 [[\u0639\u062f\u062f \u062a\u0648\u0627\u0641\u0642\u064a|\u0627\u0644\u0639\u062f\u062f \u0627\u0644\u062a\u0648\u0627\u0641\u0642\u064a]] \u0645\u0646 \u0627\u0644\u0631\u062a\u0628\u0629 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a \u0645\u0627\u0626\u0644|n}}. \u0648\u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 [[\u062a\u062d\u0644\u064a\u0644 \u0645\u0642\u0627\u0631\u0628|\u0627\u0644\u062a\u062d\u0644\u064a\u0644 \u0627\u0644\u0645\u0642\u0627\u0631\u0628]] \u0644\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u062a\u0648\u0627\u0641\u0642\u064a\u0629 \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/p>\n<p>: &lt;math&gt;<br \/>\noperatorname{E}(T)  = n cdot H_n = n log n + gamma n + frac{1}{2} + O(1\/n),<br \/>\n&lt;\/math&gt;<\/p>\n<p>\u0647\u0648 &lt;math&gt;gamma approx 0.5772156649&lt;\/math&gt; [[\u062b\u0627\u0628\u062a \u0623\u0648\u064a\u0644\u0631]].<\/p>\n<p>\u0648\u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 [[\u0645\u062a\u0628\u0627\u064a\u0646\u0629 \u0645\u0627\u0631\u0643\u0648\u0641]] \u0644\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 \u062d\u062f \u0623\u0639\u0644\u0649 \u0644\u0644\u0627\u062d\u062a\u0645\u0627\u0644 \u0627\u0644\u0645\u0637\u0644\u0648\u0628:<\/p>\n<p>: &lt;math&gt;operatorname{P}(T geq cn H_n) le frac{1}{c}.&lt;\/math&gt;<\/p>\n<p>\u064a\u0645\u0643\u0646 \u062a\u0639\u062f\u064a\u0644 \u0645\u0627 \u0633\u0628\u0642 \u062a\u0639\u062f\u064a\u0644\u064b\u0627 \u0637\u0641\u064a\u0641\u064b\u0627 \u0644\u0645\u0639\u0627\u0644\u062c\u0629 \u0627\u0644\u062d\u0627\u0644\u0629 \u0627\u0644\u062a\u064a \u062c\u064f\u0645\u0639\u062a \u0641\u064a\u0647\u0627 \u0628\u0639\u0636 \u0627\u0644\u0642\u0633\u0627\u0626\u0645 \u0645\u0633\u0628\u0642\u064b\u0627. \u0644\u064a\u0643\u0646 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a \u0645\u0627\u0626\u0644|k}} \u0639\u062f\u062f \u0642\u0633\u0627\u0626\u0645 \u0627\u0644\u0645\u062c\u0645\u0648\u0639\u0629 \u0645\u0633\u0628\u0642\u064b\u0627\u060c \u0639\u0646\u062f\u0626\u0630\u064d:<\/p>\n<p>: &lt;math&gt;<br \/>\nbegin{align}<br \/>\noperatorname{E}(T_k) &amp; {}= operatorname{E}(t_{k+1} + t_{k+2} + cdots + t_n) \\<br \/>\n&amp; {}= n cdot left(frac{1}{1} + frac{1}{2} + cdots + frac{1}{n-k}right) \\<br \/>\n&amp; {}= n cdot H_{n-k}<br \/>\nend{align}<br \/>\n&lt;\/math&gt;<\/p>\n<p>\u0648\u0639\u0646\u062f &lt;math&gt;k=0&lt;\/math&gt; \u0646\u0633\u062a\u0639\u064a\u062f \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0627\u0644\u0623\u0635\u0644\u064a\u0629 (&lt;math&gt;operatorname{E}(T) = n cdot H_n&lt;\/math&gt;)<\/p>\n<p>===\u062d\u0633\u0627\u0628 \u0627\u0644\u062a\u0628\u0627\u064a\u0646===<br \/>\n\u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0627\u0633\u062a\u0642\u0644\u0627\u0644\u064a\u0629 \u0627\u0644\u0645\u062a\u063a\u064a\u0631\u0627\u062a \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a\u0629 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a|&#8221;t&#8221;&lt;sub&gt;&#8221;i&#8221;&lt;\/sub&gt;}} \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/p>\n<p>:&lt;math&gt;<br \/>\nbegin{align}<br \/>\noperatorname{Var}(T)&amp; {}= operatorname{Var}(t_1 + cdots + t_n) \\<br \/>\n&amp; {} = operatorname{Var}(t_1) + operatorname{Var}(t_2) + cdots + operatorname{Var}(t_n) \\<br \/>\n&amp; {} = frac{1-p_1}{p_1^2} + frac{1-p_2}{p_2^2} +  cdots + frac{1-p_n}{p_n^2} \\<br \/>\n&amp; {} = left(frac{n^2}{n^2} + frac{n^2}{(n-1)^2} +  cdots + frac{n^2}{1^2}right) &#8211; left(frac{n}{n} + frac{n}{n-1} +  cdots + frac{n}{1}right) \\<br \/>\n&amp; {} = n^2 cdot left(frac{1}{1^2} + frac{1}{2^2} + cdots + frac{1}{n^2} right) &#8211; n cdot left(frac{1}{1} + frac{1}{2} + cdots + frac{1}{n} right)\\<br \/>\n&amp; {} &lt; frac{pi^2}{6} n^2<br \/>\nend{align}<br \/>\n&lt;\/math&gt;<\/p>\n<p>\u0625\u0630 \u0625\u0646 &lt;math&gt;frac{pi^2}6=frac{1}{1^2}+frac{1}{2^2}+cdots+frac{1}{n^2}+cdots&lt;\/math&gt; (\u0631\u0627\u062c\u0639 [[\u0645\u0639\u0636\u0644\u0629 \u0628\u0627\u0632\u0644]]).<\/p>\n<p>\u0648\u0644\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 \u062d\u062f \u0623\u0639\u0644\u0649 \u0644\u0644\u0627\u062d\u062a\u0645\u0627\u0644 \u0627\u0644\u0645\u0637\u0644\u0648\u0628 \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 [[\u0645\u062a\u0628\u0627\u064a\u0646\u0629 \u062a\u0634\u064a\u0628\u064a\u0634\u064a\u0641]]:<\/p>\n<p>:&lt;math&gt;operatorname{P}left(|T- n H_n| geq cnright) le frac{pi^2}{6c^2}.&lt;\/math&gt;<\/p>\n<p>=== \u0623\u0639\u062f\u0627\u062f \u0633\u062a\u064a\u0631\u0644\u0646\u063a ===<br \/>\n\u0644\u064a\u0643\u0646 \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a \u0645\u0627\u0626\u0644|X}} \u0639\u062f\u062f \u0645\u0631\u0627\u062a \u0631\u0645\u064a \u0627\u0644\u0646\u0631\u062f \u0642\u0628\u0644 \u0638\u0647\u0648\u0631 \u062c\u0645\u064a\u0639 \u0627\u0644\u0648\u062c\u0648\u0647.<\/p>\n<p>\u062a\u064f\u0639\u0631\u064e\u0651\u0641 \u0627\u0644\u0642\u0648\u0629 \u0627\u0644\u062c\u0632\u0626\u064a\u0629 \u0628\u0627\u0644\u0639\u0644\u0627\u0642\u0629 &lt;math&gt;a^{{b}}=a!left{{batop a}right}&lt;\/math&gt; \u060c\u062d\u064a\u062b &lt;math&gt;left{{batop a}right}&lt;\/math&gt; \u0647\u0648 [[\u0639\u062f\u062f \u0633\u062a\u064a\u0631\u0644\u064a\u0646\u063a|\u0639\u062f\u062f \u0633\u062a\u064a\u0631\u0644\u0646\u063a \u0645\u0646 \u0627\u0644\u0646\u0648\u0639 \u0627\u0644\u062b\u0627\u0646\u064a]].&lt;ref&gt;{{\u0627\u0633\u062a\u0634\u0647\u0627\u062f \u0628\u062f\u0648\u0631\u064a\u0629 \u0645\u062d\u0643\u0645\u0629 | \u0627\u0644\u0623\u062e\u064a\u0631 = Rus | \u0627\u0644\u0623\u0648\u0644 = Mircea Dan | \u0639\u0646\u0648\u0627\u0646 = Yet another note on notation | \u0635\u062d\u064a\u0641\u0629 = International Journal of Mathematical Education in Science and Technology | \u0646\u0627\u0634\u0631 = Informa UK Limited | \u062a\u0627\u0631\u064a\u062e = 3 Aug 2026 | issn = 0020-739X | \u062f\u0648\u064a = 10.1080\/0020739x.2026.2701739 | \u0635\u0641\u062d\u0627\u062a = 1\u201317}}&lt;\/ref&gt;<\/p>\n<p>\u062a\u062a\u0642\u0627\u0628\u0644 \u062a\u0633\u0644\u0633\u0644\u0627\u062a {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a \u0645\u0627\u0626\u0644|k}} \u0645\u0646 \u0631\u0645\u064a\u0627\u062a \u0627\u0644\u0646\u0631\u062f \u0645\u0639 \u0627\u0644\u062f\u0648\u0627\u0644 &lt;math&gt;krightarrow n&lt;\/math&gt; \u0627\u0644\u062a\u064a \u064a\u0628\u0644\u063a \u0639\u062f\u062f\u0647\u0627 &lt;math&gt;n^k&lt;\/math&gt;\u060c \u0641\u064a \u062d\u064a\u0646 \u064a\u0628\u0644\u063a \u0639\u062f\u062f \u0627\u0644\u062f\u0648\u0627\u0644 \u0627\u0644\u0634\u0627\u0645\u0644\u0629 (\u0627\u0644\u062a\u064a \u062a\u0635\u064a\u0628 \u0643\u0644 \u0648\u062c\u0647 \u0645\u0631\u0629\u064b \u0639\u0644\u0649 \u0627\u0644\u0623\u0642\u0644) &lt;math&gt;n^{{k}}&lt;\/math&gt;\u060c \u0648\u0645\u0646 \u062b\u064e\u0645\u064e\u0651 \u064a\u0643\u0648\u0646 \u0627\u062d\u062a\u0645\u0627\u0644 \u0638\u0647\u0648\u0631 \u062c\u0645\u064a\u0639 \u0627\u0644\u0648\u062c\u0648\u0647 \u0641\u064a \u063a\u0636\u0648\u0646 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a \u0645\u0627\u0626\u0644|k}} \u0631\u0645\u064a\u0627\u062a \u0647\u0648 &lt;math&gt;P(Xle k)=frac{n^{{k}}}{n^k}&lt;\/math&gt;. \u0648\u0628\u062a\u0637\u0628\u064a\u0642 \u0639\u0644\u0627\u0642\u0629 \u0627\u0644\u062a\u0643\u0631\u0627\u0631 \u0644\u0623\u0639\u062f\u0627\u062f \u0633\u062a\u064a\u0631\u0644\u0646\u063a\u060c \u064a\u0643\u0648\u0646 \u0627\u062d\u062a\u0645\u0627\u0644 \u0623\u0646 \u062a\u0643\u0641\u064a {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a \u0645\u0627\u0626\u0644|k}} \u0631\u0645\u064a\u0627\u062a \u0628\u0627\u0644\u0636\u0628\u0637 \u0647\u0648 &lt;math&gt;P(X=k)=frac{n^{{k}}}{n^k}-frac{n^{{k-1}}}{n^{k-1}}=frac{(n-1)^{{k-1}}}{n^{k-1}}&lt;\/math&gt;<\/p>\n<p>===\u0627\u0644\u062f\u0648\u0627\u0644 \u0627\u0644\u0645\u0648\u0644\u062f\u0629===<br \/>\n\u064a\u064f\u0646\u062a\u062c \u062a\u0639\u0648\u064a\u0636 &lt;math&gt;z&lt;\/math&gt; \u0628&lt;math&gt;1+z&lt;\/math&gt; \u0641\u064a \u0627\u0644\u062f\u0627\u0644\u0629 \u0627\u0644\u0645\u0648\u0644\u062f\u0629 \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u064a\u0629{{\u0645\u0644\u0627|({{\u0627\u0644\u0644\u063a\u0629|en|Probability generating function}})}} \u062f\u0627\u0644\u0629 \u0645\u0648\u0644\u062f\u0629 \u0639\u0627\u062f\u064a\u0629{{\u0645\u0644\u0627|({{\u0627\u0644\u0644\u063a\u0629|en|ordinary generating function}}) &lt;ref group=&#8221;\u0639\u0631&#8221;&gt;{{\u0627\u0633\u062a\u0634\u0647\u0627\u062f \u0628\u0648\u064a\u0643\u064a \u0628\u064a\u0627\u0646\u0627\u062a|Q108593221|\u0635=495}}&lt;\/ref&gt;}} \u0644&lt;math&gt;Eleft[{Xchoose k}right]&lt;\/math&gt;. \u0648\u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 [[\u062a\u062d\u0644\u064a\u0644 \u0627\u0644\u0643\u0633\u0648\u0631 \u0627\u0644\u062c\u0632\u0626\u064a\u0629]] &lt;math&gt;{frac1x-1choose n}^{-1}=sum_{k=0}^n{nchoose k}frac{(-1)^{n-k}}{1-kx}&lt;\/math&gt;\u060c \u064a\u0645\u0643\u0646 \u0623\u062e\u0630 \u0627\u0644\u062a\u0648\u0633\u0639:<br \/>\n:&lt;math&gt;begin{aligned}&amp;{frac n{x+1}choose n}^{-1}\\<br \/>\n=&amp;sum_{i=0}^n{nchoose i}frac{(-1)^{n-i}}{1-i(1-frac n{x+1+n})}\\<br \/>\n=&amp;sum_{i=0}^n{nchoose i}(-1)^{n-i}left(frac{1+n}{1+n-i}+insum_{k=1}^inftyfrac{(i-1)^{k-1}}{(n+1-i)^{k+1}}x^kright)end{aligned}&lt;\/math&gt;<\/p>\n<p>\u0644\u0630\u0644\u0643\u060c \u0641\u0645\u0646 \u0645\u0646 \u0623\u062c\u0644 &lt;math&gt;k&gt;0&lt;\/math&gt;:<br \/>\n:&lt;math&gt;Eleft[{Xchoose k}right]=nsum_{i=0}^n{nchoose i}(-1)^{n-i}ifrac{(i-1)^{k-1}}{(n+1-i)^{k+1}}&lt;\/math&gt;<br \/>\n\u0628\u0627\u0644\u0646\u0633\u0628\u0629 \u0625\u0644\u0649 \u062f\u0627\u0644\u0629 \u0645\u0648\u0644\u062f\u0629 \u0639\u0627\u062f\u064a\u0629 {{\u062a\u0639\u0628\u064a\u0631 \u0631\u064a\u0627\u0636\u064a \u0645\u0627\u0626\u0644|f}}\u060c \u0648\u0644\u0623\u0646 &lt;math&gt;left(frac x{1-x}right)^i=sum_{n=0}^infty{k-1choose i-1}x^k&lt;\/math&gt;\u060c \u0641\u0625\u0646 \u062a\u063a\u064a\u064a\u0631\u064b\u0627{{\u0645\u0644\u0627|({{\u0627\u0644\u0644\u063a\u0629|en|variation}})}} \u0645\u0646 [[\u062a\u062d\u0648\u064a\u0644 \u062b\u0646\u0627\u0626\u064a|\u0627\u0644\u062a\u062d\u0648\u064a\u0644 \u0627\u0644\u062b\u0646\u0627\u0626\u064a]] \u064a\u0646\u062a\u062c &lt;math&gt;[x^k]fleft(frac x{1+x}right)=sum_{i=0}^k{k-1choose i-1}(-1)^{k-i}[x^i]f(x)&lt;\/math&gt;. (\u0648\u062a\u062d\u062f\u064a\u062f\u0627\u064b\u060c \u0625\u0630\u0627 \u0643\u0627\u0646&lt;math&gt;{frac n{x+1}choose n}^{-1}=fleft(frac x{1+x}right)&lt;\/math&gt;, &lt;math&gt;f(x)={n-nxchoose n}^{-1}&lt;\/math&gt;.)<\/p>\n<p>\u0648\u0628\u0625\u0639\u0627\u062f\u0629 \u0643\u062a\u0627\u0628\u0629 \u0645\u0639\u0627\u0645\u0644 \u0627\u0644\u062a\u062d\u0648\u064a\u0644 \u0627\u0644\u062b\u0646\u0627\u0626\u064a \u0639\u0628\u0631 [[\u062f\u0627\u0644\u0629 \u063a\u0627\u0645\u0627]] \u0648\u062a\u0648\u0633\u064a\u0639\u0647 \u0639\u0644\u0649 \u0647\u064a\u0626\u0629  &lt;math&gt;exp&lt;\/math&gt; \u0644\u0645\u062a\u0633\u0644\u0633\u0644\u0629 {{\u0648\u0625\u0648|\u062f\u0627\u0644\u0629 \u063a\u0627\u0645\u0627 \u0627\u0644\u0645\u062a\u0639\u062f\u062f\u0629|Polygamma function}} (\u0628\u062f\u0644\u0627\u0644\u0629 [[\u0639\u062f\u062f \u062a\u0648\u0627\u0641\u0642\u064a|\u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u062a\u0648\u0627\u0641\u0642\u064a\u0629 \u0627\u0644\u0645\u0639\u0645\u0645\u0629]])\u060c \u0646\u062c\u062f \u0623\u0646\u064e\u0651:<br \/>\n:&lt;math&gt;left[frac{x^i}{i!}right]{n-xchoose n}^{-1}=sum_{Pinmathrm{perms}(i)}prod_{cin P}H^{(|c|)}_n&lt;\/math&gt;<br \/>\n\u0648\u0645\u0646 \u062b\u064e\u0645\u064e\u0651:<br \/>\n:&lt;math&gt;Eleft[{Xchoose k}right]=sum_{i=0}^k{k-1choose i-1}(-1)^{k-i}frac{n^i}{i!}sum_{Pinmathrm{perms}(i)}prod_{cin P}H^{(|c|)}_n&lt;\/math&gt;<\/p>\n<p>\u0648\u064a\u0645\u0643\u0646 \u0623\u064a\u0636\u0627\u064b \u0643\u062a\u0627\u0628\u0629 \u0647\u0630\u0647 \u0627\u0644\u0635\u064a\u063a\u0629 \u0628\u062f\u0644\u0627\u0644\u0629 \u0627\u0644\u062d\u062f\u0648\u062f\u064a\u0627\u062a \u0627\u0644\u0647\u0627\u0628\u0637\u0629 \u0648\u0639\u062f\u062f \u0644\u0627\u0647{{\u0645\u0644\u0627|({{\u0627\u0644\u0644\u063a\u0629|en|Lah number}}) (\u0623\u064a \u0627\u0644\u0623\u0639\u062f\u0627\u062f \u0627\u0644\u0645\u0624\u0634\u0631\u0629 \u0648\u063a\u064a\u0631 \u0627\u0644\u0645\u0624\u0634\u0631\u0629)}} \u0639\u0644\u0649 \u0627\u0644\u0646\u062d\u0648 \u0627\u0644\u0622\u062a\u064a:<br \/>\n:&lt;math&gt;E[x^underline k]=sum_{i=0}^kL(k,i)(-1)^{k-i}n^isum_{Pinmathrm{perms}(i)}prod_{cin P}H^{(|c|)}_n&lt;\/math&gt;<br \/>\n\u0648\u064a\u0645\u0643\u0646 \u0627\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 [[\u0639\u0632\u0645 (\u0631\u064a\u0627\u0636\u064a\u0627\u062a)|\u0627\u0644\u0639\u0632\u0648\u0645 \u0627\u0644\u062e\u0627\u0645]] \u0644\u0644\u062a\u0648\u0632\u064a\u0639 \u0645\u0646 \u0627\u0644\u0639\u0632\u0648\u0645 \u0627\u0644\u0647\u0627\u0628\u0637\u0629 \u0639\u0628\u0631 \u062a\u062d\u0648\u064a\u0644 \u0633\u062a\u064a\u0631\u0644\u0646\u063a{{\u0645\u0644\u0627|({{\u0627\u0644\u0644\u063a\u0629|en|Stirling transform}})}}\u061b \u0648\u0627\u0633\u062a\u0646\u0627\u062f\u0627\u064b \u0625\u0644\u0649 \u0627\u0644\u0645\u062a\u0637\u0627\u0628\u0642\u0629:<br \/>\n: &lt;math&gt;left{{Katop i}right}(-1)^K=sum_{k=0}^Kleft{{Katop k}right}L(k,i)(-1)^k&lt;\/math&gt;<br \/>\n\u064a\u0646\u062a\u062c:<br \/>\n:&lt;math&gt;E[x^k]=sum_{i=0}^kleft{{katop i}right}(-1)^{k-i}n^i!!sum_{Pinmathrm{perms}(i)}prod_{cin P}H^{(|c|)}_n&lt;\/math&gt;<\/p>\n<p>==\u062a\u0642\u062f\u064a\u0631\u0627\u062a \u0627\u0644\u0630\u064a\u0644==<\/p>\n<p>\u064a\u0645\u0643\u0646 \u0627\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 \u062a\u0642\u062f\u064a\u0631 \u0623\u0642\u0648\u0649 \u0644\u0630\u064a\u0644 \u0645\u0646 \u0627\u0644\u062c\u0647\u0629 \u0627\u0644\u0639\u0644\u064a\u0627 \u0639\u0644\u0649 \u0627\u0644\u0646\u062d\u0648 \u0627\u0644\u0622\u062a\u064a\u060c  \u0646\u0631\u0645\u0632 \u0628\u0627\u0644\u0631\u0645\u0632 &lt;math&gt;{Z}_i^r&lt;\/math&gt;  [[\u062d\u062f\u062b (\u0646\u0638\u0631\u064a\u0629 \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u0627\u062a)|\u0644\u0644\u062d\u062f\u062b]] \u0627\u0644\u0630\u064a \u064a\u0639\u0646\u064a \u0639\u062f\u0645 \u0627\u062e\u062a\u064a\u0627\u0631 \u0627\u0644\u0642\u0633\u064a\u0645\u0629 &lt;math&gt;i&lt;\/math&gt; \u0641\u064a \u0623\u0648\u0644 &lt;math&gt;r&lt;\/math&gt; \u0633\u062d\u0628\u0627\u062a. \u0639\u0646\u062f\u0626\u0630\u064d:<br \/>\n&lt;math&gt;r = beta n log n&lt;\/math&gt;\u060c \u064a\u0635\u0628\u062d &lt;math&gt;Pleft [ {Z}_i^r right ] le e^{(-beta n log n ) \/ n} = n^{-beta}&lt;\/math&gt;.<br \/>\n\u0648\u0644\u0630\u0644\u0643\u060c \u0645\u0646 \u0623\u062c\u0644: &lt;math&gt;r = beta n log n&lt;\/math&gt;, we have &lt;math&gt;Pleft [ {Z}_i^r right ] le e^{(-beta n log n ) \/ n} = n^{-beta}&lt;\/math&gt;<\/p>\n<p>\u0648\u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0645\u062a\u0628\u0627\u064a\u0646\u0629 \u0628\u0648\u0644 {{\u0645\u0644\u0627|({{\u0627\u0644\u0644\u063a\u0629|en|Boole&#8217;s inequality}})}} \u0639\u0644\u0649 &lt;math&gt;n&lt;\/math&gt; \u0642\u0633\u064a\u0645\u0629\u060c \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<br \/>\n:&lt;math&gt;<br \/>\nbegin{align}<br \/>\nPleft [ T &gt; beta n log n right ] = P left [ \tbigcup_i {Z}_i^{beta n log n} right ] le n cdot P [ {Z}_1^{beta n log n} ] le n^{-beta + 1}.<br \/>\nend{align}<br \/>\n&lt;\/math&gt;<\/p>\n<p>== \u0627\u0646\u0638\u0631 \u0623\u064a\u0636\u064b\u0627 ==<br \/>\n* [[\u0645\u0639\u0636\u0644\u0629 \u064a\u0648\u0645 \u0627\u0644\u0645\u064a\u0644\u0627\u062f]]<\/p>\n<p>==\u0627\u0644\u0647\u0648\u0627\u0645\u0634 ==<br \/>\n{{\u0645\u0644\u0627\u062d\u0638\u0627\u062a}}<\/p>\n<p>==\u0627\u0644\u0645\u0631\u0627\u062c\u0639==<br \/>\n===\u0627\u0644\u0627\u0633\u062a\u0634\u0647\u0627\u062f===<br \/>\n; \u0627\u0644\u0645\u0631\u0627\u062c\u0639 \u0627\u0644\u0639\u0631\u0628\u064a\u0629<br \/>\n{{\u0645\u0631\u0627\u062c\u0639|\u0645\u062c\u0645\u0648\u0639\u0629=\u0639\u0631|2}}<\/p>\n<p>; \u0627\u0644\u0645\u0631\u0627\u062c\u0639 \u0627\u0644\u0623\u062c\u0646\u0628\u064a\u0629<br \/>\n{{\u0645\u0631\u0627\u062c\u0639|30em|2}}<\/p>\n<p>{{\u0634\u0631\u064a\u0637 \u0633\u0641\u0644\u064a \u0645\u0633\u0627\u0626\u0644 \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u0627\u062a}}<br \/>\n{{\u0634\u0631\u064a\u0637 \u0628\u0648\u0627\u0628\u0627\u062a|\u0625\u062d\u0635\u0627\u0621|\u0627\u0644\u0627\u062d\u0635\u0627\u0621 \u0648\u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644|\u062a\u062d\u0644\u064a\u0644 \u0631\u064a\u0627\u0636\u064a|\u0631\u064a\u0627\u0636\u064a\u0627\u062a}}<\/p>\n<p>[[\u062a\u0635\u0646\u064a\u0641:\u0645\u0633\u0627\u0626\u0644 \u0631\u064a\u0627\u0636\u064a\u0627\u062a]]<br \/>\n[[\u062a\u0635\u0646\u064a\u0641:\u0645\u0628\u0631\u0647\u0646\u0627\u062a \u0641\u064a \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u0627\u062a]]<\/div>\n<\/div>\n<div class=\"pvc_clear\"><\/div>\n<p id=\"pvc_stats_2520\" class=\"pvc_stats total_only  \" data-element-id=\"2520\" style=\"\"><i class=\"pvc-stats-icon medium\" aria-hidden=\"true\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" version=\"1.0\" viewBox=\"0 0 502 315\" preserveAspectRatio=\"xMidYMid meet\"><g transform=\"translate(0,332) scale(0.1,-0.1)\" fill=\"\" stroke=\"none\"><path d=\"M2394 3279 l-29 -30 -3 -207 c-2 -182 0 -211 15 -242 39 -76 157 -76 196 0 15 31 17 60 15 243 l-3 209 -33 29 c-26 23 -41 29 -80 29 -41 0 -53 -5 -78 -31z\"\/><path d=\"M3085 3251 c-45 -19 -58 -50 -96 -229 -47 -217 -49 -260 -13 -295 52 -53 146 -42 177 20 16 31 87 366 87 410 0 70 -86 122 -155 94z\"\/><path d=\"M1751 3234 c-13 -9 -29 -31 -37 -50 -12 -29 -10 -49 21 -204 19 -94 39 -189 45 -210 14 -50 54 -80 110 -80 34 0 48 6 76 34 21 21 34 44 34 59 0 14 -18 113 -40 219 -37 178 -43 195 -70 221 -36 32 -101 37 -139 11z\"\/><path d=\"M1163 3073 c-36 -7 -73 -59 -73 -102 0 -56 133 -378 171 -413 34 -32 83 -37 129 -13 70 36 67 87 -16 290 -86 209 -89 214 -129 231 -35 14 -42 15 -82 7z\"\/><path d=\"M3689 3066 c-15 -9 -33 -30 -42 -48 -48 -103 -147 -355 -147 -375 0 -98 131 -148 192 -74 13 15 57 108 97 206 80 196 84 226 37 273 -30 30 -99 39 -137 18z\"\/><path d=\"M583 2784 c-38 -19 -67 -74 -58 -113 9 -42 211 -354 242 -373 16 -10 45 -18 66 -18 51 0 107 52 107 100 0 39 -1 41 -124 234 -80 126 -108 162 -133 173 -41 17 -61 16 -100 -3z\"\/><path d=\"M4250 2784 c-14 -9 -74 -91 -133 -183 -95 -150 -107 -173 -107 -213 0 -55 33 -94 87 -104 67 -13 90 8 211 198 130 202 137 225 78 284 -27 27 -42 34 -72 34 -22 0 -50 -8 -64 -16z\"\/><path d=\"M2275 2693 c-553 -48 -1095 -270 -1585 -649 -135 -104 -459 -423 -483 -476 -23 -49 -22 -139 2 -186 73 -142 361 -457 571 -626 285 -228 642 -407 990 -497 242 -63 336 -73 660 -74 310 0 370 5 595 52 535 111 1045 392 1455 803 122 121 250 273 275 326 19 41 19 137 0 174 -41 79 -309 363 -465 492 -447 370 -946 591 -1479 653 -113 14 -422 18 -536 8z m395 -428 c171 -34 330 -124 456 -258 112 -119 167 -219 211 -378 27 -96 24 -300 -5 -401 -72 -255 -236 -447 -474 -557 -132 -62 -201 -76 -368 -76 -167 0 -236 14 -368 76 -213 98 -373 271 -451 485 -162 444 86 934 547 1084 153 49 292 57 452 25z m909 -232 c222 -123 408 -262 593 -441 76 -74 138 -139 138 -144 0 -16 -233 -242 -330 -319 -155 -123 -309 -223 -461 -299 l-81 -41 32 46 c18 26 49 83 70 128 143 306 141 649 -6 957 -25 52 -61 116 -79 142 l-34 47 45 -20 c26 -10 76 -36 113 -56z m-2057 25 c-40 -58 -105 -190 -130 -263 -110 -324 -59 -707 132 -981 25 -35 42 -64 37 -64 -19 0 -241 119 -326 174 -188 122 -406 314 -532 468 l-58 71 108 103 c185 178 428 349 672 473 66 33 121 60 123 61 2 0 -10 -19 -26 -42z\"\/><path d=\"M2375 1950 c-198 -44 -350 -190 -395 -379 -18 -76 -8 -221 19 -290 114 -284 457 -406 731 -260 98 52 188 154 231 260 27 69 37 214 19 290 -38 163 -166 304 -326 360 -67 23 -215 33 -279 19z\"\/><\/g><\/svg><\/i> <img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"16\" height=\"16\" alt=\"Loading\" src=\"https:\/\/i0.wp.com\/dev95.site\/wp-content\/plugins\/page-views-count\/ajax-loader-2x.gif?resize=16%2C16&#038;ssl=1\" border=0 \/><\/p>\n<div class=\"pvc_clear\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>SanBonne: {{\u0635\u0646\u062f\u0648\u0642 \u0645\u0639\u0644\u0648\u0645\u0627\u062a \u062a\u0648\u0632\u064a\u0639 \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u0627\u062a | name = \u0645\u0633\u0623\u0644\u0629 \u062c\u0627\u0645\u0639 \u0627\u0644\u0642\u0633\u0627\u0626\u0645 | type = \u0645\u0646\u0641\u0635\u0644 | parameters = &lt;math&gt;ninmathbb N&lt;\/math&gt; &amp;ndash; \u0639\u062f\u062f \u0623\u0648\u062c\u0647 \u0627\u0644\u0646\u0631\u062f | support = &lt;math&gt;kinmathbb N&lt;\/math&gt; &amp;ndash; \u0627\u0644\u0623\u062f\u0648\u0627\u0631 \u0627\u0644\u0644\u0627\u0632\u0645\u0629 \u0644\u062a\u0638\u0647\u0631 \u0627\u0644\u0623\u0648\u062c\u0647 \u062c\u0645\u064a\u0639\u0647\u0627 &lt;!&#8211; | pdf = &lt;math&gt;frac{(n-1)^{{k-1}}}{n^{k-1}}&lt;\/math&gt;<\/p>\n<div class=\"hosteria-entry-more\"><a href=\"https:\/\/dev95.site\/ar\/%d9%85%d8%b3%d8%a3%d9%84%d8%a9-%d8%ac%d8%a7%d9%85%d8%b9-%d8%a7%d9%84%d9%82%d8%b3%d8%a7%d8%a6%d9%85\/\" class=\"no-underline font-light  group-hover:text-primary-800 dark:group-hover:text-primary-300 py-1\">Read more &gt;&gt;&gt;<\/a><\/div>\n<div class=\"pvc_clear\"><\/div>\n<p id=\"pvc_stats_2520\" class=\"pvc_stats total_only\" data-element-id=\"2520\" style=\"\"><i class=\"pvc-stats-icon medium\" aria-hidden=\"true\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" version=\"1.0\" viewbox=\"0 0 502 315\" preserveaspectratio=\"xMidYMid meet\"><g transform=\"translate(0,332) scale(0.1,-0.1)\" fill=\"\" stroke=\"none\"><path d=\"M2394 3279 l-29 -30 -3 -207 c-2 -182 0 -211 15 -242 39 -76 157 -76 196 0 15 31 17 60 15 243 l-3 209 -33 29 c-26 23 -41 29 -80 29 -41 0 -53 -5 -78 -31z\"\/><path d=\"M3085 3251 c-45 -19 -58 -50 -96 -229 -47 -217 -49 -260 -13 -295 52 -53 146 -42 177 20 16 31 87 366 87 410 0 70 -86 122 -155 94z\"\/><path d=\"M1751 3234 c-13 -9 -29 -31 -37 -50 -12 -29 -10 -49 21 -204 19 -94 39 -189 45 -210 14 -50 54 -80 110 -80 34 0 48 6 76 34 21 21 34 44 34 59 0 14 -18 113 -40 219 -37 178 -43 195 -70 221 -36 32 -101 37 -139 11z\"\/><path d=\"M1163 3073 c-36 -7 -73 -59 -73 -102 0 -56 133 -378 171 -413 34 -32 83 -37 129 -13 70 36 67 87 -16 290 -86 209 -89 214 -129 231 -35 14 -42 15 -82 7z\"\/><path d=\"M3689 3066 c-15 -9 -33 -30 -42 -48 -48 -103 -147 -355 -147 -375 0 -98 131 -148 192 -74 13 15 57 108 97 206 80 196 84 226 37 273 -30 30 -99 39 -137 18z\"\/><path d=\"M583 2784 c-38 -19 -67 -74 -58 -113 9 -42 211 -354 242 -373 16 -10 45 -18 66 -18 51 0 107 52 107 100 0 39 -1 41 -124 234 -80 126 -108 162 -133 173 -41 17 -61 16 -100 -3z\"\/><path d=\"M4250 2784 c-14 -9 -74 -91 -133 -183 -95 -150 -107 -173 -107 -213 0 -55 33 -94 87 -104 67 -13 90 8 211 198 130 202 137 225 78 284 -27 27 -42 34 -72 34 -22 0 -50 -8 -64 -16z\"\/><path d=\"M2275 2693 c-553 -48 -1095 -270 -1585 -649 -135 -104 -459 -423 -483 -476 -23 -49 -22 -139 2 -186 73 -142 361 -457 571 -626 285 -228 642 -407 990 -497 242 -63 336 -73 660 -74 310 0 370 5 595 52 535 111 1045 392 1455 803 122 121 250 273 275 326 19 41 19 137 0 174 -41 79 -309 363 -465 492 -447 370 -946 591 -1479 653 -113 14 -422 18 -536 8z m395 -428 c171 -34 330 -124 456 -258 112 -119 167 -219 211 -378 27 -96 24 -300 -5 -401 -72 -255 -236 -447 -474 -557 -132 -62 -201 -76 -368 -76 -167 0 -236 14 -368 76 -213 98 -373 271 -451 485 -162 444 86 934 547 1084 153 49 292 57 452 25z m909 -232 c222 -123 408 -262 593 -441 76 -74 138 -139 138 -144 0 -16 -233 -242 -330 -319 -155 -123 -309 -223 -461 -299 l-81 -41 32 46 c18 26 49 83 70 128 143 306 141 649 -6 957 -25 52 -61 116 -79 142 l-34 47 45 -20 c26 -10 76 -36 113 -56z m-2057 25 c-40 -58 -105 -190 -130 -263 -110 -324 -59 -707 132 -981 25 -35 42 -64 37 -64 -19 0 -241 119 -326 174 -188 122 -406 314 -532 468 l-58 71 108 103 c185 178 428 349 672 473 66 33 121 60 123 61 2 0 -10 -19 -26 -42z\"\/><path d=\"M2375 1950 c-198 -44 -350 -190 -395 -379 -18 -76 -8 -221 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