{"id":26800,"date":"2021-06-13T13:00:19","date_gmt":"2021-06-13T13:00:19","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=26800"},"modified":"2024-05-22T03:46:30","modified_gmt":"2024-05-22T03:46:30","slug":"%d8%a7%d9%84%d9%85%d8%aa%d8%ba%d9%8a%d8%b1%d8%a7%d8%aa-%d8%a7%d9%84%d8%b9%d8%b4%d9%88%d8%a7%d8%a6%d9%8a%d8%a9-%d9%88%d8%aa%d9%88%d8%b2%d9%8a%d8%b9%d8%a7%d8%aa-%d8%a7%d9%84%d8%a7%d8%ad%d8%aa%d9%85","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/ar\/%d8%a7%d9%84%d9%85%d8%aa%d8%ba%d9%8a%d8%b1%d8%a7%d8%aa-%d8%a7%d9%84%d8%b9%d8%b4%d9%88%d8%a7%d8%a6%d9%8a%d8%a9-%d9%88%d8%aa%d9%88%d8%b2%d9%8a%d8%b9%d8%a7%d8%aa-%d8%a7%d9%84%d8%a7%d8%ad%d8%aa%d9%85\/","title":{"rendered":"\u0627\u0644\u0645\u062a\u063a\u064a\u0631\u0627\u062a \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a\u0629 \u0648\u062a\u0648\u0632\u064a\u0639\u0627\u062a \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; 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\u0627\u0644\u0648\u0636\u0639 \u0627\u0644\u062d\u0627\u0644\u064a \u0628\u062f\u0627\u0626\u064a\u0629 \u0646\u0648\u0639\u064b\u0627 \u0645\u0627\u061b \u0627\u0644\u0645\u062a\u063a\u064a\u0631\u0627\u062a \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a\u0629 \u0648\u062a\u0648\u0632\u064a\u0639\u0627\u062a \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644 \u0647\u064a\u060c \u0625\u0646 \u0635\u062d \u0627\u0644\u0642\u0648\u0644\u060c \u0645\u0641\u0627\u0647\u064a\u0645 \u062a\u062a\u064a\u062d \u0644\u0646\u0627 \u00ab\u062a\u062d\u0633\u064a\u0646 \u0642\u062f\u0631\u062a\u0646\u0627 \u0639\u0644\u0649 \u0627\u0644\u0639\u0645\u0644 \u0645\u0639 \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u0627\u062a \u0648\u0625\u062c\u0631\u0627\u0621 \u0627\u0644\u062a\u062d\u0644\u064a\u0644 \u0627\u0644\u0625\u062d\u0635\u0627\u0626\u064a\u00bb.<\/p>\n<p><a name=\"1\"><\/a><\/br><\/br><\/p>\n<h2>\u0645\u0627 \u0647\u064a \u0627\u0644\u0645\u062a\u063a\u064a\u0631\u0627\u062a \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a\u0629\u061f<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><strong><span style=\"color: #ff0000;\">\u0644\u062a\u0639\u0631\u064a\u0641 \u0645\u0641\u0647\u0648\u0645 \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a<\/span><\/strong>\u060c \u0645\u0646 \u0627\u0644\u0645\u0641\u064a\u062f \u0627\u0644\u0628\u062f\u0621 \u0628\u0646\u0647\u062c \u0628\u062f\u064a\u0647\u064a: \u064a\u0645\u0643\u0646 \u062a\u0641\u0633\u064a\u0631 \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a \u0643\u0640 \u00ab\u0631\u0642\u0645 \u064a\u0639\u062a\u0645\u062f \u0639\u0644\u0649 \u0646\u062a\u064a\u062c\u0629 \u062a\u062c\u0631\u0628\u0629 \u0639\u0634\u0648\u0627\u0626\u064a\u0629\u00bb. \u0648\u0645\u0639 \u0630\u0644\u0643\u060c \u0644\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 \u0641\u0647\u0645 \u0623\u062f\u0642\u060c \u0645\u0646 \u0627\u0644\u0636\u0631\u0648\u0631\u064a \u0623\u064a\u0636\u064b\u0627 \u0627\u0633\u062a\u0643\u0634\u0627\u0641 \u062a\u0639\u0631\u064a\u0641\u0647 \u0627\u0644\u0631\u0633\u0645\u064a. \u062f\u0639\u0648\u0646\u0627 \u0646\u0631\u0649 \u0647\u0630\u0627 \u0627\u0644\u062a\u0639\u0631\u064a\u0641:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0ffe0;\">\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000080;\"><strong>\u062a\u0639\u0631\u064a\u0641:<\/strong><\/span> \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a \u0639\u0644\u0649 \u0645\u062c\u0645\u0648\u0639\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{X}<\/span><\/span> \u0647\u0648 \u062f\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f:\\Omega \\longmapsto \\mathcal{X}<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">\u0627\u0644\u062d\u0627\u0644\u0629 \u0627\u0644\u0623\u0643\u062b\u0631 \u0634\u064a\u0648\u0639\u064b\u0627 \u0647\u064a \u0639\u0646\u062f\u0645\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{X}= \\mathbb{R},<\/span><\/span> \u0648\u060c \u0645\u0627 \u0644\u0645 \u064a\u064f\u0630\u0643\u0631 \u062e\u0644\u0627\u0641 \u0630\u0644\u0643\u060c \u0647\u0630\u0627 \u0647\u0648 \u0645\u0627 \u0633\u0646\u0641\u062a\u0631\u0636\u0647 \u0645\u0646 \u0627\u0644\u0622\u0646 \u0641\u0635\u0627\u0639\u062f\u064b\u0627\u061b \u0623\u064a \u0623\u0646\u0646\u0627 \u0633\u0646\u0639\u0645\u0644 \u0645\u0639 \u0645\u062a\u063a\u064a\u0631\u0627\u062a \u0639\u0634\u0648\u0627\u0626\u064a\u0629 \u0630\u0627\u062a \u0642\u064a\u0645 \u062d\u0642\u064a\u0642\u064a\u0629. \u0639\u0627\u062f\u0629\u064b \u0645\u0627 \u064a\u062a\u0645 \u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0627\u0644\u0623\u062d\u0631\u0641 \u0627\u0644\u0643\u0628\u064a\u0631\u0629 \u0644\u0644\u062f\u0644\u0627\u0644\u0629 \u0639\u0644\u0649 \u0627\u0644\u0645\u062a\u063a\u064a\u0631\u0627\u062a \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a\u0629\u060c \u0645\u062b\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X,Y,Z, \\cdots,<\/span><\/span>\u060c \u0628\u064a\u0646\u0645\u0627 \u062a\u064f\u0633\u062a\u062e\u062f\u0645 \u0627\u0644\u0623\u062d\u0631\u0641 \u0627\u0644\u0635\u063a\u064a\u0631\u0629 \u0644\u0644\u062f\u0644\u0627\u0644\u0629 \u0639\u0644\u0649 \u0627\u0644\u062b\u0648\u0627\u0628\u062a. \u0644\u062a\u0628\u0633\u064a\u0637 \u0627\u0644\u0623\u0645\u0648\u0631\u060c \u0633\u0646\u0634\u064a\u0631 \u0625\u0644\u0649 \u0627\u0644\u0645\u062a\u063a\u064a\u0631\u0627\u062a \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a\u0629 \u0628\u0628\u0633\u0627\u0637\u0629 \u0639\u0644\u0649 \u0623\u0646\u0647\u0627 \u00ab\u0645\u062a\u063a\u064a\u0631\u0627\u062a\u00bb.<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0e0ff;\">\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000000;\"><strong>\u0645\u062b\u0627\u0644:<\/strong><\/span> \u0644\u0646\u0641\u062a\u0631\u0636 \u0623\u0646 \u0646\u0631\u0645\u064a \u0646\u0631\u062f \u0630\u0648 6 \u0623\u0648\u062c\u0647 \u0645\u0631\u062a\u064a\u0646. \u0641\u064a \u0647\u0630\u0647 \u0627\u0644\u062d\u0627\u0644\u0629 \u0633\u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2d6} = \\{(\\omega_1, \\omega_2)\\;|\\; \\omega_1,\\omega_2 \\in \\{1,2,3,4,5,6\\}\\}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0645\u0646 \u0647\u0630\u0627\u060c \u064a\u0645\u0643\u0646\u0646\u0627 \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0645\u062a\u063a\u064a\u0631\u0627\u062a \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a\u0629 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=<\/span><\/span> \u00ab\u0639\u062f\u062f \u0645\u0631\u0627\u062a \u0627\u0644\u062d\u0635\u0648\u0644 \u0639\u0644\u0649 \u0627\u0644\u0631\u0642\u0645 \u0648\u0627\u062d\u062f\u00bb<\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y=<\/span><\/span> \u00ab\u0645\u062c\u0645\u0648\u0639 \u0627\u0644\u0646\u062a\u0627\u0626\u062c\u00bb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">=\\omega_1 + \\omega_2<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Z=<\/span><\/span> \u00ab\u0646\u062a\u064a\u062c\u0629 \u0627\u0644\u0631\u0645\u064a\u0629 \u0627\u0644\u062b\u0627\u0646\u064a\u0629\u00bb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">= \\omega_2<\/span><\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"2\"><\/a><\/br><\/br><\/p>\n<h2>\u0645\u0627 \u0647\u064a \u062a\u0648\u0632\u064a\u0639\u0627\u062a \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u061f<\/h2>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0ffe0;\">\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000080;\"><strong>\u062a\u0639\u0631\u064a\u0641: <\/strong><\/span><strong><span style=\"color: #ff0000;\">\u062f\u0627\u0644\u0629 \u0627\u0644\u062a\u0648\u0632\u064a\u0639 (\u0623\u0648 \u00abFD\u00bb)<\/span><\/strong> \u0644\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u0647\u064a \u062f\u0627\u0644\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X: \\mathbb{R} \\longmapsto \\mathbb{R}<\/span><\/span> \u0645\u0639\u0631\u0641\u0629 \u0628\u0627\u0644\u0639\u0644\u0627\u0642\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X(x) = P(\\{\\omega \\;|\\; X(\\omega)\\leq x\\}),<\/span><\/span> \u0623\u0648 \u0628\u0627\u062e\u062a\u0635\u0627\u0631: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X\\leq x).<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">\u0628\u0634\u0643\u0644 \u0639\u0627\u0645\u060c \u0645\u0627 \u064a\u0647\u0645 \u0641\u064a \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a \u0644\u064a\u0633 \u062a\u0639\u0628\u064a\u0631\u0647 \u0627\u0644\u0635\u0631\u064a\u062d \u0641\u064a \u0641\u0636\u0627\u0621 \u0627\u0644\u0639\u064a\u0646\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span><\/span>\u060c \u0628\u0644 \u062f\u0627\u0644\u0629 \u062a\u0648\u0632\u064a\u0639\u0647. \u064a\u0645\u0643\u0646 \u062d\u0630\u0641 \u0627\u0644\u0645\u0624\u0634\u0631 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u0641\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X<\/span><\/span> \u0625\u0630\u0627 \u0643\u0627\u0646 \u0627\u0644\u0633\u064a\u0627\u0642 \u0648\u0627\u0636\u062d\u064b\u0627 \u0648\u0644\u0627 \u064a\u0648\u062c\u062f \u063a\u0645\u0648\u0636. \u0645\u0646 \u0627\u0644\u0634\u0627\u0626\u0639 \u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u0627\u0644\u0631\u0645\u0648\u0632 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim F<\/span><\/span> \u0644\u0644\u0625\u0634\u0627\u0631\u0629 \u0625\u0644\u0649 \u0623\u0646 \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u0644\u062f\u064a\u0647 \u062f\u0627\u0644\u0629 \u062a\u0648\u0632\u064a\u0639 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F.<\/span><\/span><\/p>\n<p><a name=\"3\"><\/a><\/br><\/br><\/p>\n<h3>\u062e\u0635\u0627\u0626\u0635 \u062a\u0648\u0632\u064a\u0639\u0627\u062a \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=gIKn9t1hnrw&amp;t=806s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0625\u0630\u0627 \u0643\u0627\u0646\u062a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> \u062f\u0627\u0644\u0629 \u062a\u0648\u0632\u064a\u0639 \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u0627\u062a<\/span><\/strong><\/a> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b<\/span><\/span> \u0647\u0645\u0627 \u0623\u064a \u0639\u062f\u062f\u064a\u0646 \u062d\u0642\u064a\u0642\u064a\u064a\u0646\u060c \u0641\u0633\u062a\u062a\u062d\u0642\u0642 \u0627\u0644\u062e\u0635\u0627\u0626\u0635 \u0627\u0644\u062a\u0627\u0644\u064a\u0629:<\/p>\n<p style=\"text-align: justify; color: #000000;\">(a) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b \\longrightarrow [P(a\\lt X \\leq b) = F(b) - F(a)]<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">(b) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b \\longrightarrow F(a) \\leq F(b),<\/span><\/span> \u0623\u064a \u0623\u0646 \u00abF \u062a\u0632\u0627\u064a\u062f\u064a\u0629\u00bb.<\/p>\n<p style=\"text-align: justify; color: #000000;\">(c) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to +\\infty} F(x) = 1<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to -\\infty} F(x) = 0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">(d) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(X=x)=\\lim_{t\\to x^+}F(t) - \\lim_{t\\to x^-}F(t)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">(e) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle F(x)=\\lim_{t\\to x^+}F(t)<\/span><\/span><\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td style=\"background-color: #ffe0e0;\"><span class=\"collapseomatic \" id=\"id69e43360d2202\"  tabindex=\"0\" title=\"\u0628\u0631\u0647\u0627\u0646\"    >\u0628\u0631\u0647\u0627\u0646<\/span><div id=\"target-id69e43360d2202\" class=\"collapseomatic_content \">\n<p style=\"text-align: justify; color: #000000;\"><strong>(a)<\/strong> \u0644\u062a\u0643\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B<\/span><\/span> \u0647\u0645\u0627 \u0627\u0644\u0623\u062d\u062f\u0627\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{X\\leq a\\}<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{X\\leq b\\}<\/span><\/span> \u0639\u0644\u0649 \u0627\u0644\u062a\u0648\u0627\u0644\u064a\u060c \u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b.<\/span><\/span> \u0625\u0630\u0627 \u062a\u062d\u0642\u0642 \u0643\u0644 \u0647\u0630\u0627\u060c \u0641\u0625\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A\\subseteq B<\/span><\/span> \u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{blue}{P(a\\lt X\\leq b)} = P(B\\setminus A) = P(B) - P(B\\cap A) = P(B)-P(A) =\\color{blue}{F(b) - F(a)}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\"><strong>(b)<\/strong> \u0645\u0646 \u0627\u0644\u062c\u0632\u0621 (a) \u064a\u062a\u0628\u064a\u0646: \u0628\u0645\u0627 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(B\\setminus A)\\geq 0,<\/span><\/span> \u0641\u0625\u0646:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(b) - F(a) \\geq 0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0648\u0647\u0648 \u0646\u0641\u0633 \u0627\u0644\u0634\u064a\u0621 \u0628\u0642\u0648\u0644\u0646\u0627:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(a) \\leq F(b)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\"><strong>(c)<\/strong> \u0647\u0646\u0627 \u0633\u0646\u0633\u062a\u062e\u062f\u0645 \u062d\u0642\u064a\u0642\u0629 \u0623\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> \u062f\u0627\u0644\u0629 \u062a\u0632\u0627\u064a\u062f\u064a\u0629 (\u062b\u0628\u062a \u0630\u0644\u0643 \u0641\u064a (b)) \u0648\u0645\u062d\u062f\u062f\u0629 \u0628\u0623\u0642\u0635\u0649 \u0642\u064a\u0645\u0629 \u00ab1\u00bb (\u0644\u0623\u0646 \u0627\u0644\u062a\u0648\u0632\u064a\u0639 \u064a\u0639\u0631\u0641 \u0645\u0646 \u062d\u064a\u062b \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644). \u0641\u0642\u0637 \u0628\u0647\u0630\u0627 \u064a\u0643\u0641\u064a \u0623\u0646 \u0646\u0642\u0648\u0644:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty} F(x) = 1<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0646\u0647\u062c \u062a\u0643\u0645\u064a\u0644\u064a \u0644\u0647\u0630\u0627 \u064a\u0633\u0645\u062d \u0644\u0646\u0627 \u0628\u0625\u062c\u0631\u0627\u0621 \u0627\u0644\u062d\u0633\u0627\u0628\u0627\u062a \u0627\u0644\u062a\u0627\u0644\u064a\u0629 \u0628\u0646\u0641\u0633 \u0627\u0644\u0646\u062a\u064a\u062c\u0629.<\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0644\u0646\u062d\u062f\u062f \u0627\u0644\u0645\u062c\u0645\u0648\u0639\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A_n=\\{\\omega\\;|\\;X(\\omega)\\leq n\\}.<\/span><\/span> \u0645\u0646 \u0647\u0630\u0627 \u0633\u0647\u0644 \u0627\u0644\u062a\u062d\u0642\u0642 \u0623\u0646\u0647\u060c \u0644\u0643\u0644 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> \u064a\u062d\u062f\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A_{n}\\subseteq A_{n+1},<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\bigcup_{n\\lt +\\infty} A_n = \\Omega<\/span><\/span> \u0648\u0628\u0627\u0644\u062a\u0627\u0644\u064a\u060c \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 \u062e\u0627\u0635\u064a\u0629 <a href=\"http:\/\/toposuranos.com\/material\/ar\/%d8%a7%d9%84%d9%86%d8%b8%d8%b1%d9%8a%d8%a7%d8%aa-%d8%a7%d9%84%d9%85%d9%81%d9%8a%d8%af%d8%a9-%d9%84%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d8%a7%d8%ad%d8%aa%d9%85%d8%a7%d9%84%d8%a7%d8%aa\/\" rel=\"noopener\" target=\"_blank\">\u0627\u0644\u0627\u0633\u062a\u0645\u0631\u0627\u0631\u064a\u0629<\/a> \u0633\u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle 1=P(\\Omega) = P\\left( \\bigcup_{n\\lt +\\infty} A_n \\right) = \\lim_{n\\to +\\infty} P(A_n) = \\lim_{n\\to +\\infty} P(\\{\\omega\\;|\\;X(\\omega)\\leq n\\}) = \\lim_{n\\to +\\infty} P(X\\leq n)=\\lim_{n\\to +\\infty}F(n)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0623\u064a:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\lim_{x\\to +\\infty} F(x) = 1}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0628\u0627\u0644\u0645\u0642\u0627\u0628\u0644\u060c \u0644\u0644\u062d\u062f \u062d\u064a\u062b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to -\\infty<\/span><\/span>\u060c \u0644\u062f\u064a\u0646\u0627 \u0645\u0627 \u064a\u0644\u064a:<\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0623\u0648\u0644\u0627\u064b \u0644\u0646\u062d\u062f\u062f \u0627\u0644\u0645\u062c\u0645\u0648\u0639\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B_n=\\{\\omega\\;|\\;-n\\lt X(\\omega)\\}.<\/span><\/span> \u0645\u0646 \u0647\u0630\u0627 \u064a\u062a\u062d\u0642\u0642:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{n \\to -\\infty}F(n) = \\lim_{n\\to -\\infty} P(X\\leq n) = \\lim_{n\\to \\infty} P(X\\leq -n)= 1 - \\lim_{n\\to \\infty} P(-n \\lt X) = 1 - \\lim_{n\\to \\infty}P(B_n)) = 1 - P(\\Omega) = 1-1=0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\"><strong>(d)<\/strong> \u064a\u062a\u0628\u0639 \u0646\u0641\u0633 \u0645\u0646\u0637\u0642 \u0627\u0644\u062c\u0632\u0621 (c). \u0646\u0628\u062f\u0623 \u0628\u062a\u062d\u062f\u064a\u062f \u0627\u0644\u0645\u062c\u0645\u0648\u0639\u0629<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle C_n = \\left\\{x - \\frac{1}{n} \\leq X \\leq x + \\frac{1}{n}\\right\\}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0645\u0646 \u0647\u0630\u0627 \u0644\u062f\u064a\u0646\u0627<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> C_{n+1}\\subseteq C_n<\/span><\/span><\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcap_{n\\gt 0} C_n = \\{X=x\\}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0644\u0630\u0644\u0643\u060c \u0628\u0627\u0633\u062a\u062e\u062f\u0627\u0645 <a href=\"http:\/\/toposuranos.com\/material\/ar\/%d8%a7%d9%84%d9%86%d8%b8%d8%b1%d9%8a%d8%a7%d8%aa-%d8%a7%d9%84%d9%85%d9%81%d9%8a%d8%af%d8%a9-%d9%84%d8%ad%d8%b3%d8%a7%d8%a8-%d8%a7%d9%84%d8%a7%d8%ad%d8%aa%d9%85%d8%a7%d9%84%d8%a7%d8%aa\/\" rel=\"noopener\" target=\"_blank\">\u0646\u062a\u064a\u062c\u0629 \u062e\u0627\u0635\u064a\u0629 \u0627\u0644\u0627\u0633\u062a\u0645\u0631\u0627\u0631\u064a\u0629<\/a> \u0646\u062d\u0635\u0644 \u0639\u0644\u0649:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(X=x)=P\\left(\\bigcap_{n\\gt 0} C_n \\right) = \\lim_{n\\to \\infty} P(C_n) = \\lim_{x+1\/n \\to x^+}F\\left(x+1\/n\\right) - \\lim_{x-1\/n \\to x^-}F\\left(x-1\/n\\right)= \\lim_{t \\to x^+}F\\left(t\\right) - \\lim_{t \\to x^-}F\\left(t\\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\"><strong>(e)<\/strong> \u0627\u0644\u062d\u0627\u0644\u0629 \u0627\u0644\u0623\u062e\u064a\u0631\u0629 \u062a\u0623\u062a\u064a \u0645\u0646 \u0627\u0644\u0646\u062a\u064a\u062c\u0629 \u0627\u0644\u0633\u0627\u0628\u0642\u0629. \u0641\u064a \u0627\u0644\u0648\u0627\u0642\u0639\u060c \u0628\u0645\u0627 \u0623\u0646\u0646\u0627 \u0623\u062b\u0628\u062a\u0646\u0627<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(X=x)= \\lim_{t \\to x^+}F\\left(t\\right) - \\lim_{t \\to x^-}F\\left(t\\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u064a\u0645\u0643\u0646\u0646\u0627 \u0623\u0646 \u0646\u0643\u062a\u0628<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{t \\to x^+}F\\left(t\\right) = P(X=x) + \\lim_{t \\to x^-}F\\left(t\\right) = P(X=x) + \\lim_{t\\to x^-}P(X\\leq t)= P(X\\leq x) = F(x)<\/span><\/span><\/p>\n<\/div><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"4\"><\/a><\/br><\/br><\/p>\n<h3>\u0627\u0644\u0639\u0644\u0627\u0642\u0629 \u0628\u064a\u0646 \u0627\u0644\u0645\u062a\u063a\u064a\u0631\u0627\u062a \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a\u0629 \u0648\u062a\u0648\u0632\u064a\u0639\u0627\u062a \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644<\/h3>\n<p style=\"text-align: justify; color: #000000;\">\u064a\u0642\u0627\u0644 \u0625\u0646 \u0645\u062a\u063a\u064a\u0631\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y<\/span><\/span> \u0644\u0647\u0645\u0627 \u0646\u0641\u0633 \u062a\u0648\u0632\u064a\u0639 \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644 \u0625\u0630\u0627 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall A\\subseteq \\mathbb{R})(P(X\\in A) = P(Y\\in A)).<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u064a\u0645\u0643\u0646 \u0623\u0646 \u064a\u0643\u0648\u0646 \u0644\u0645\u062a\u063a\u064a\u0631\u064a\u0646 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y<\/span><\/span> \u0645\u0639\u0631\u0641\u064a\u0646 \u0639\u0644\u0649 \u0646\u0641\u0633 \u0641\u0636\u0627\u0621 \u0627\u0644\u0639\u064a\u0646\u0629 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span><\/span> \u0646\u0641\u0633 \u0627\u0644\u062a\u0648\u0632\u064a\u0639 \u062f\u0648\u0646 \u0623\u0646 \u064a\u0643\u0648\u0646\u0627 \u0646\u0641\u0633 \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a. \u0639\u0644\u0649 \u0633\u0628\u064a\u0644 \u0627\u0644\u0645\u062b\u0627\u0644\u060c \u0625\u0630\u0627 \u0627\u0639\u062a\u0628\u0631\u0646\u0627 \u062a\u062c\u0631\u0628\u0629 \u0631\u0645\u064a \u0639\u0645\u0644\u0629 \u0645\u062a\u0648\u0627\u0632\u0646\u0629 \u0630\u0627\u062a \u0648\u062c\u0647\u064a\u0646 \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=1<\/span><\/span> \u064a\u0645\u062b\u0644 \u0627\u0644\u0648\u062c\u0647 \u0648 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=0<\/span><\/span> \u064a\u0645\u062b\u0644 \u0627\u0644\u0646\u0642\u0634\u060c \u064a\u0645\u0643\u0646 \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y=1-X<\/span><\/span> \u0648\u0633\u0646\u062d\u0635\u0644 \u0639\u0644\u0649 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X=1) = P(Y=1)=0.5,<\/span><\/span> \u0648\u0623\u0646 \u0643\u0644\u0627\u0647\u0645\u0627 \u0644\u0647\u0645\u0627 \u0646\u0641\u0633 \u0627\u0644\u062a\u0648\u0632\u064a\u0639\u060c \u0648\u0644\u0643\u0646 \u0625\u0630\u0627 \u062d\u0633\u0628\u0646\u0627 \u0627\u062d\u062a\u0645\u0627\u0644 \u0623\u0646 \u064a\u0643\u0648\u0646 \u0644\u0647\u0645\u0627 \u0646\u0641\u0633 \u0627\u0644\u0642\u064a\u0645\u0629 \u0633\u0646\u062d\u0635\u0644 \u0639\u0644\u0649 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X=Y)=0<\/span><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0627\u0644\u0645\u062a\u063a\u064a\u0631\u0627\u062a \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a\u0629 \u0648\u062a\u0648\u0632\u064a\u0639\u0627\u062a \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644 \u0645\u0644\u062e\u0635\u062a\u0642\u062f\u0645 \u0647\u0630\u0647 \u0627\u0644\u062d\u0635\u0629 \u063a\u0648\u0635\u064b\u0627 \u0639\u0645\u064a\u0642\u064b\u0627 \u0641\u064a \u0645\u0641\u0627\u0647\u064a\u0645 \u0627\u0644\u0645\u062a\u063a\u064a\u0631\u0627\u062a \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a\u0629 \u0648\u062a\u0648\u0632\u064a\u0639\u0627\u062a \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u060c \u0648\u0647\u064a \u0623\u0639\u0645\u062f\u0629 \u0623\u0633\u0627\u0633\u064a\u0629 \u0641\u064a \u0646\u0638\u0631\u064a\u0629 \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u0627\u062a \u0648\u0627\u0644\u062a\u062d\u0644\u064a\u0644 \u0627\u0644\u0625\u062d\u0635\u0627\u0626\u064a. \u064a\u062a\u0645 \u062a\u0642\u062f\u064a\u0645 \u062a\u0639\u0631\u064a\u0641 \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a \u0643\u0631\u0642\u0645 \u064a\u0639\u062a\u0645\u062f \u0639\u0644\u0649 \u0646\u062a\u064a\u062c\u0629 \u062a\u062c\u0631\u0628\u0629 \u0639\u0634\u0648\u0627\u0626\u064a\u0629. \u064a\u062a\u0645 \u062a\u0646\u0627\u0648\u0644 \u062f\u0627\u0644\u0629 \u062a\u0648\u0632\u064a\u0639 \u0627\u0644\u0645\u062a\u063a\u064a\u0631 \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a\u060c \u0645\u0639 \u0625\u0628\u0631\u0627\u0632 \u0623\u0647\u0645\u064a\u062a\u0647\u0627 \u0648\u0643\u0630\u0644\u0643 \u062e\u0635\u0627\u0626\u0635\u0647\u0627 \u0627\u0644\u0623\u0633\u0627\u0633\u064a\u0629. \u0623\u062e\u064a\u0631\u064b\u0627\u060c \u064a\u062a\u0645 \u062a\u062d\u0644\u064a\u0644 \u0627\u0644\u0639\u0644\u0627\u0642\u0629 \u0628\u064a\u0646 \u0627\u0644\u0645\u062a\u063a\u064a\u0631\u0627\u062a \u0627\u0644\u0639\u0634\u0648\u0627\u0626\u064a\u0629 \u0648\u062a\u0648\u0632\u064a\u0639\u0627\u062a \u0627\u0644\u0627\u062d\u062a\u0645\u0627\u0644\u060c 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