{"id":27277,"date":"2021-10-08T13:00:09","date_gmt":"2021-10-08T13:00:09","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27277"},"modified":"2024-06-20T11:32:01","modified_gmt":"2024-06-20T11:32:01","slug":"%e0%a4%a8%e0%a4%bf%e0%a4%b0%e0%a4%82%e0%a4%a4%e0%a4%b0-%e0%a4%b8%e0%a4%82%e0%a4%ad%e0%a4%be%e0%a4%b5%e0%a5%8d%e0%a4%af%e0%a4%a4%e0%a4%be-%e0%a4%b5%e0%a4%bf%e0%a4%a4%e0%a4%b0%e0%a4%a3","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/hi\/%e0%a4%a8%e0%a4%bf%e0%a4%b0%e0%a4%82%e0%a4%a4%e0%a4%b0-%e0%a4%b8%e0%a4%82%e0%a4%ad%e0%a4%be%e0%a4%b5%e0%a5%8d%e0%a4%af%e0%a4%a4%e0%a4%be-%e0%a4%b5%e0%a4%bf%e0%a4%a4%e0%a4%b0%e0%a4%a3\/","title":{"rendered":"\u0928\u093f\u0930\u0902\u0924\u0930 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0935\u093f\u0924\u0930\u0923"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>\u0928\u093f\u0930\u0902\u0924\u0930 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0935\u093f\u0924\u0930\u0923<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>\u0938\u093e\u0930\u093e\u0902\u0936<\/strong><br \/><em>\u092f\u0939\u093e\u0902 \u0939\u092e \u0928\u093f\u0930\u0902\u0924\u0930 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0935\u093f\u0924\u0930\u0923 \u0915\u0940 \u0905\u0935\u0927\u093e\u0930\u0923\u093e \u0915\u0940 \u0917\u0939\u0930\u093e\u0908 \u0938\u0947 \u091c\u093e\u0902\u091a \u0915\u0930\u0947\u0902\u0917\u0947, \u092a\u093e\u0902\u091a \u0938\u092c\u0938\u0947 \u092a\u094d\u0930\u0938\u093f\u0926\u094d\u0927 \u0935\u093f\u0924\u0930\u0923\u094b\u0902 \u0915\u0940 \u0935\u093f\u0936\u0947\u0937\u0924\u093e\u0913\u0902 \u0914\u0930 \u0909\u092a\u092f\u094b\u0917\u094b\u0902 \u092a\u0930 \u092a\u094d\u0930\u0915\u093e\u0936 \u0921\u093e\u0932\u0924\u0947 \u0939\u0941\u090f: \u090f\u0915\u094d\u0938\u092a\u094b\u0928\u0947\u0936\u093f\u092f\u0932 \u0935\u093f\u0924\u0930\u0923, \u0938\u092e\u093e\u0928 \u0906\u092f\u0924\u093e\u0915\u093e\u0930 \u0935\u093f\u0924\u0930\u0923, \u0938\u093e\u092e\u093e\u0928\u094d\u092f (\u0917\u0949\u0938\u093f\u092f\u0928) \u0935\u093f\u0924\u0930\u0923, \u0935\u0947\u0907\u092c\u0941\u0932 \u0935\u093f\u0924\u0930\u0923 \u0914\u0930 \u0917\u093e\u092e\u093e \u0935\u093f\u0924\u0930\u0923\u0964 \u0907\u0928 \u0935\u093f\u0924\u0930\u0923\u094b\u0902 \u092e\u0947\u0902 \u0938\u0947 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0915\u094b \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0928\u0947 \u0935\u093e\u0932\u0947 \u0917\u0923\u093f\u0924\u0940\u092f \u0938\u0942\u0924\u094d\u0930 \u092a\u094d\u0930\u0926\u093e\u0928 \u0915\u093f\u090f \u0917\u090f \u0939\u0948\u0902, \u0914\u0930 \u0907\u0928\u0915\u0947 \u0928\u093f\u0939\u093f\u0924\u093e\u0930\u094d\u0925\u094b\u0902 \u0914\u0930 \u0935\u094d\u092f\u093e\u0935\u0939\u093e\u0930\u093f\u0915 \u0905\u0928\u0941\u092a\u094d\u0930\u092f\u094b\u0917\u094b\u0902 \u0915\u0940 \u091c\u093e\u0902\u091a \u0915\u0940 \u0917\u0908 \u0939\u0948, \u091c\u0948\u0938\u0947 \u0915\u093f \u0930\u0947\u0921\u093f\u092f\u094b\u0927\u0930\u094d\u092e\u0940 \u0928\u092e\u0942\u0928\u094b\u0902 \u092e\u0947\u0902 \u0915\u0923\u094b\u0902 \u0915\u0947 \u0909\u0924\u094d\u0938\u0930\u094d\u091c\u0928 \u0915\u093e \u092e\u0942\u0932\u094d\u092f\u093e\u0902\u0915\u0928 \u092f\u093e \u090f\u0915 \u0930\u0947\u0932 \u092e\u0947\u0902 \u090f\u0915 \u0917\u0947\u0902\u0926 \u0915\u0940 \u0938\u094d\u0925\u093f\u0924\u093f \u0915\u0940 \u0917\u0923\u0928\u093e\u0964 \u0907\u0938\u0915\u0947 \u0905\u0932\u093e\u0935\u093e, \u092f\u0939 \u0935\u093f\u0938\u094d\u0924\u093e\u0930 \u0938\u0947 \u092c\u0924\u093e\u092f\u093e \u0917\u092f\u093e \u0939\u0948 \u0915\u093f \u0907\u0928 \u0935\u093f\u0924\u0930\u0923\u094b\u0902 \u0915\u094b \u0935\u093f\u0936\u093f\u0937\u094d\u091f \u092e\u093e\u092a\u0926\u0902\u0921\u094b\u0902 \u0915\u0947 \u0906\u0935\u0947\u0926\u0928 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u0915\u0948\u0938\u0947 \u0938\u0902\u0936\u094b\u0927\u093f\u0924 \u0914\u0930 \u0905\u0928\u0941\u0915\u0942\u0932\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>\u0905\u0927\u093f\u0917\u092e \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f:<\/strong><br \/>\n\u0907\u0938 \u0915\u0915\u094d\u0937\u093e \u0915\u0947 \u0905\u0902\u0924 \u092e\u0947\u0902, \u091b\u093e\u0924\u094d\u0930 \u0938\u0915\u094d\u0937\u092e \u0939\u094b\u0902\u0917\u0947:\n<\/p>\n<ol>\n<li><strong>\u0938\u092e\u091d\u0928\u093e<\/strong> \u0915\u093f \u0928\u093f\u0930\u0902\u0924\u0930 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0935\u093f\u0924\u0930\u0923 \u0915\u094d\u092f\u093e \u0939\u0948\u0902\u0964<\/li>\n<li><strong>\u0932\u093e\u0917\u0942 \u0915\u0930\u0928\u093e<\/strong> \u0938\u092c\u0938\u0947 \u092a\u094d\u0930\u0938\u093f\u0926\u094d\u0927 \u0928\u093f\u0930\u0902\u0924\u0930 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0935\u093f\u0924\u0930\u0923: \u090f\u0915\u094d\u0938\u092a\u094b\u0928\u0947\u0936\u093f\u092f\u0932, \u0938\u092e\u093e\u0928 \u0906\u092f\u0924\u093e\u0915\u093e\u0930, \u090f\u0915\u094d\u0938\u092a\u094b\u0928\u0947\u0936\u093f\u092f\u0932, \u0938\u093e\u092e\u093e\u0928\u094d\u092f (\u0917\u0949\u0938\u093f\u092f\u0928), \u0935\u0947\u0907\u092c\u0941\u0932, \u0914\u0930 \u0917\u093e\u092e\u093e\u0964<\/li>\n<\/ol>\n<p><center><br \/>\n<strong><u>\u0938\u093e\u092e\u0917\u094d\u0930\u0940 \u0915\u093e \u0938\u0942\u091a\u0915\u093e\u0902\u0915<\/u>:<\/strong><br \/>\n<a href=\"#1\"><strong>\u0928\u093f\u0930\u0902\u0924\u0930 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0935\u093f\u0924\u0930\u0923 \u0915\u094d\u092f\u093e \u0939\u0948\u0902?<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>\u092a\u093e\u0902\u091a \u0938\u092c\u0938\u0947 \u092a\u094d\u0930\u0938\u093f\u0926\u094d\u0927 \u0928\u093f\u0930\u0902\u0924\u0930 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0935\u093f\u0924\u0930\u0923<\/strong><\/a><br \/>\n<a href=\"#3\">\u090f\u0915\u094d\u0938\u092a\u094b\u0928\u0947\u0936\u093f\u092f\u0932 \u0935\u093f\u0924\u0930\u0923<\/a><br \/>\n<a href=\"#4\">\u0938\u092e\u093e\u0928 \u0906\u092f\u0924\u093e\u0915\u093e\u0930 \u0935\u093f\u0924\u0930\u0923<\/a><br \/>\n<a href=\"#5\">\u0938\u093e\u092e\u093e\u0928\u094d\u092f (\u0917\u0949\u0938\u093f\u092f\u0928) \u0935\u093f\u0924\u0930\u0923<\/a><br \/>\n<a href=\"#6\">\u0935\u0947\u0907\u092c\u0941\u0932 \u0935\u093f\u0924\u0930\u0923<\/a><br \/>\n<a href=\"#7\">\u0917\u093e\u092e\u093e \u0935\u093f\u0924\u0930\u0923<\/a><br \/>\n<a href=\"#8\"><strong>\u0905\u092d\u094d\u092f\u093e\u0938<\/strong><\/a><br \/>\n<\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/REOTUa7K8uQ\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p style=\"text-align: justify; color: #000000;\">\u091c\u092c \u0939\u092e\u0928\u0947 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0915\u0947 <a href=\"http:\/\/toposuranos.com\/material\/es\/conoce-el-espacio-muestral-de-la-teoria-de-las-probabilidades\/\" target=\"_blank\" rel=\"noopener\">\u0928\u092e\u0942\u0928\u093e \u0938\u094d\u0925\u093e\u0928\u094b\u0902<\/a> \u0915\u0947 \u092c\u093e\u0930\u0947 \u092e\u0947\u0902 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0940, \u0924\u094b \u0939\u092e\u0928\u0947 \u0926\u0947\u0916\u093e \u0915\u093f \u092f\u0947 \u0926\u094b \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0947 \u0939\u094b \u0938\u0915\u0924\u0947 \u0939\u0948\u0902: \u0915\u0941\u091b \u0935\u093f\u0935\u093f\u0915\u094d\u0924 \u0914\u0930 \u0905\u0928\u094d\u092f \u0928\u093f\u0930\u0902\u0924\u0930\u0964 \u0939\u092e\u0928\u0947 \u092f\u0939 \u092d\u0940 \u0938\u092e\u0940\u0915\u094d\u0937\u093e \u0915\u0940 \u0915\u093f \u090f\u0915 <a href=\"http:\/\/toposuranos.com\/material\/es\/distribuciones-discretas-de-probabilidad-y-ejemplos\/\" target=\"_blank\" rel=\"noopener\">\u0935\u093f\u0935\u093f\u0915\u094d\u0924 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0935\u093f\u0924\u0930\u0923<\/a> \u0915\u094d\u092f\u093e \u0939\u094b\u0924\u093e \u0939\u0948\u0964 \u0905\u092c \u092f\u0939 \u0928\u093f\u0930\u0902\u0924\u0930 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0935\u093f\u0924\u0930\u0923 \u0915\u0940 \u092c\u093e\u0930\u0940 \u0939\u0948\u0964<\/p>\n<p>&nbsp;<\/p>\n<p><a name=\"1\"><\/a><\/br><\/br><\/p>\n<h2>\u0928\u093f\u0930\u0902\u0924\u0930 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0935\u093f\u0924\u0930\u0923 \u0915\u094d\u092f\u093e \u0939\u0948\u0902?<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=86s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0939\u092e \u0915\u0939\u0947\u0902\u0917\u0947 \u0915\u093f \u090f\u0915 \u092f\u093e\u0926\u0943\u091a\u094d\u091b\u093f\u0915 \u091a\u0930<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u0915\u093e \u090f\u0915 \u0928\u093f\u0930\u0902\u0924\u0930 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0935\u093f\u0924\u0930\u0923 \u0939\u0948 \u092f\u0926\u093f \u090f\u0915 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f_X : \\mathbb{R} \\longrightarrow \\mathbb{R}^+,<\/span><\/span> \u0928\u093e\u092e\u0915 \u090f\u0915 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u092e\u094c\u091c\u0942\u0926 \u0939\u0948 \u091c\u093f\u0938\u0947 \u0939\u092e <strong><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u0915\u093e \u0918\u0928\u0924\u094d\u0935<\/strong> \u0915\u0939\u0947\u0902\u0917\u0947, \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\forall A \\subseteq \\mathbb{R}<\/span><\/span> \u0915\u0947 \u0932\u093f\u090f \u092f\u0939 \u0938\u092e\u093e\u0928\u0924\u093e \u092e\u093e\u0928\u094d\u092f \u0939\u094b\u0917\u0940<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X\\in A) = \\displaystyle \\int_A f_X(x)dx<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0935\u093f\u0936\u0947\u0937 \u0930\u0942\u092a \u0938\u0947, \u092f\u0926\u093f \u0939\u092e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A=]a,b]<\/span><\/span> \u0932\u0947\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0939\u092e\u0947\u0902 \u092e\u093f\u0932\u0947\u0917\u093e<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(a\\lt X \\leq b) = \\displaystyle \\int_a^b f_X(x)dx<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0914\u0930 \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a=-\\infty<\/span><\/span><\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X(x) = P( X \\leq x) = \\displaystyle \\int_{-\\infty}^x f_X(t)dt<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0914\u0930 \u0907\u0938\u0915\u0947 \u0905\u0932\u093e\u0935\u093e, \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0935\u093f\u0924\u0930\u0923 \u0915\u0940 \u0938\u0902\u092a\u0924\u094d\u0924\u093f (c) (<a href=\"http:\/\/toposuranos.com\/material\/es\/variables-aleatorias-y-distribuciones-de-probabilidades\/\" rel=\"noopener\" target=\"_blank\">\u092f\u0939\u093e\u0902 \u0926\u0947\u0916\u0947\u0902<\/a>) \u0938\u0947 \u0939\u092e\u0947\u0902 \u092e\u093f\u0932\u0947\u0917\u093e<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int_{-\\infty}^{+\\infty} f_X(t)dt = 1<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0907\u0938 \u0905\u0902\u0924\u093f\u092e \u0905\u092d\u093f\u0935\u094d\u092f\u0915\u094d\u0924\u093f \u092a\u0930 \u092e\u094c\u0932\u093f\u0915 \u092a\u094d\u0930\u092e\u0947\u092f \u0932\u093e\u0917\u0942 \u0915\u0930\u0928\u0947 \u0938\u0947 \u0939\u092e\u0947\u0902 \u092e\u093f\u0932\u0947\u0917\u093e \u0915\u093f \u090f\u0915 \u0928\u093f\u0930\u0902\u0924\u0930 \u0935\u093f\u0924\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X(x),<\/span><\/span> \u0938\u092d\u0940 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u0915\u0947 \u0932\u093f\u090f \u0928\u093f\u0930\u0902\u0924\u0930 \u0939\u0948, \u0914\u0930 \u0907\u0938\u0915\u093e \u0905\u0935\u0915\u0932\u091c \u0938\u092d\u0940 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> \u092e\u093e\u0928\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f_X(x)<\/span><\/span> \u0939\u0948 \u091c\u0939\u093e\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f_X(x)<\/span><\/span> \u0928\u093f\u0930\u0902\u0924\u0930 \u0939\u0948\u0964 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X(x)<\/span><\/span> \u0915\u0940 \u0928\u093f\u0930\u0902\u0924\u0930\u0924\u093e \u0914\u0930 \u0938\u0902\u092a\u0924\u094d\u0924\u093f (d) (<a href=\"http:\/\/toposuranos.com\/material\/es\/variables-aleatorias-y-distribuciones-de-probabilidades\/\" rel=\"noopener\" target=\"_blank\">\u092f\u0939\u093e\u0902 \u0926\u0947\u0916\u0947\u0902<\/a>) \u0938\u0947 \u0939\u092e\u0947\u0902 \u092e\u093f\u0932\u0947\u0917\u093e:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(x=X)=0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0914\u0930 \u0907\u0938\u0932\u093f\u090f<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(x\\leq X)= P(x\\lt X)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f<\/span><\/span> \u0915\u094b\u0908 \u092d\u0940 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0939\u0948 \u091c\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f\\geq 0<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int_{-\\infty}^{+\\infty}f(x)dx = 1,<\/span><\/span> \u0924\u094b \u0907\u0938\u0947 \u090f\u0915 \u0918\u0928\u0924\u094d\u0935 \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/p>\n<p><a name=\"2\"><\/a><\/br><\/br><\/p>\n<h2>\u092a\u093e\u0902\u091a \u0938\u092c\u0938\u0947 \u092a\u094d\u0930\u0938\u093f\u0926\u094d\u0927 \u0928\u093f\u0930\u0902\u0924\u0930 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0935\u093f\u0924\u0930\u0923<\/h2>\n<p><a name=\"3\"><\/a><\/br><\/br><\/p>\n<h3>\u090f\u0915\u094d\u0938\u092a\u094b\u0928\u0947\u0936\u093f\u092f\u0932 \u0935\u093f\u0924\u0930\u0923<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=714s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u090f\u0915 \u090f\u0915\u094d\u0938\u092a\u094b\u0928\u0947\u0936\u093f\u092f\u0932 \u0935\u093f\u0924\u0930\u0923 \u092b\u093c\u0902\u0915\u094d\u0936\u0928<\/span><\/strong><\/a> \u091c\u093f\u0938\u092e\u0947\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha \\gt 0 <\/span><\/span> \u0915\u093e \u092e\u093e\u0928 \u0939\u0948, \u0907\u0938 \u0930\u0942\u092a \u092e\u0947\u0902 \u090f\u0915 \u0935\u093f\u0924\u0930\u0923 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0939\u0948:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(t) = \\left\\{\\begin{array}{lll}\n\n1 - e^{-t\/\\alpha} &amp; ; &amp; t\\geq 0 \\\\ \\\\\n\n0 &amp; ; &amp; t\\lt 0\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0928\u0924\u0940\u091c\u0924\u0928, \u0907\u0938\u0915\u093e \u0918\u0928\u0924\u094d\u0935 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0939\u0948<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f(t) = \\left\\{\\begin{array}{lll}\n\n\\frac{1}{\\alpha}e^{-t\/\\alpha} &amp; ; &amp; t\\geq 0 \\\\ \\\\\n\n0 &amp; ; &amp; t\\lt 0\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u092f\u0926\u093f \u090f\u0915 \u092f\u093e\u0926\u0943\u091a\u094d\u091b\u093f\u0915 \u091a\u0930 \u092e\u0947\u0902 \u090f\u0915\u094d\u0938\u092a\u094b\u0928\u0947\u0936\u093f\u092f\u0932 \u0935\u093f\u0924\u0930\u0923 \u0939\u0948, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> \u0915\u0947 \u092e\u093e\u0928 \u0915\u0947 \u0938\u093e\u0925, \u0939\u092e \u0932\u093f\u0916\u0924\u0947 \u0939\u0948\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim Ex(\\alpha).<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u092a\u0949\u0907\u0938\u0928 \u0935\u093f\u0924\u0930\u0923 \u0915\u0947 \u0938\u0902\u0926\u0930\u094d\u092d \u092e\u0947\u0902, \u092f\u0926\u093f \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938 \u090f\u0915 \u0930\u0947\u0921\u093f\u092f\u094b\u0927\u0930\u094d\u092e\u0940 \u0928\u092e\u0942\u0928\u093e \u0939\u0948 \u091c\u094b \u0914\u0938\u0924 \u0909\u0924\u094d\u0938\u0930\u094d\u091c\u0928 \u0926\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c,<\/span><\/span> \u0915\u0947 \u0938\u093e\u0925 \u090f\u0915 \u0915\u0923 \u0915\u093e \u0909\u0924\u094d\u0938\u0930\u094d\u091c\u0928 \u0915\u0930\u0924\u093e \u0939\u0948, \u0924\u094b \u092a\u0939\u0932\u093e \u0915\u0923 \u0909\u0924\u094d\u092a\u0928\u094d\u0928 \u0939\u094b\u0928\u0947 \u0915\u093e \u0938\u092e\u092f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T<\/span><\/span> \u0915\u0947 \u0938\u093e\u0925 \u090f\u0915 \u090f\u0915\u094d\u0938\u092a\u094b\u0928\u0947\u0936\u093f\u092f\u0932 \u0935\u093f\u0924\u0930\u0923 \u0939\u094b\u0917\u093e, \u091c\u093f\u0938\u0915\u093e \u092e\u093e\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1\/c<\/span><\/span> \u0939\u0948\u0964 \u0926\u0942\u0938\u0930\u0947 \u0936\u092c\u094d\u0926\u094b\u0902 \u092e\u0947\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T\\sim Ex(1\/c),<\/span><\/span> \u0914\u0930 \u092a\u0930\u093f\u0923\u093e\u092e\u0938\u094d\u0935\u0930\u0942\u092a:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(T\\geq t)= e^{-ct}<\/span><\/span><\/p>\n<p><a name=\"4\"><\/a><\/br><\/br><\/p>\n<h3>\u0938\u092e\u093e\u0928 \u0906\u092f\u0924\u093e\u0915\u093e\u0930 \u0935\u093f\u0924\u0930\u0923<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=930s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u090f\u0915 \u0938\u092e\u093e\u0928 \u0906\u092f\u0924\u093e\u0915\u093e\u0930 \u0935\u093f\u0924\u0930\u0923<\/span><\/strong><\/a> \u090f\u0915 \u0905\u0902\u0924\u0930\u093e\u0932 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b]<\/span><\/span> \u092a\u0930 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948, \u0907\u0938 \u0918\u0928\u0924\u094d\u0935 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0926\u094d\u0935\u093e\u0930\u093e \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0939\u094b\u0924\u093e \u0939\u0948:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\left\\{\\begin{array}{lll}\n\n\\displaystyle\\frac{1}{b-a} &amp; ; &amp; x\\in[a,b] \\\\ \\\\\n\n0 &amp; ; &amp; E.O.C.\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u092f\u0926\u093f \u0939\u092e \u090f\u0915 \u091b\u094b\u091f\u0940 \u0917\u0947\u0902\u0926 \u0915\u094b \u090f\u0915 \u0930\u0947\u0932 \u092e\u0947\u0902 \u091b\u094b\u0921\u093c\u0924\u0947 \u0939\u0948\u0902 \u091c\u093f\u0938\u0915\u0947 \u0938\u093f\u0930\u094b\u0902 \u092a\u0930 \u0938\u0940\u092e\u093e \u0939\u0948 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[a,b],<\/span><\/span> \u0914\u0930 \u092f\u0939 \u0909\u091b\u0932\u0924\u0940 \u0939\u0948, \u0924\u094b <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u092f\u093e\u0926\u0943\u091a\u094d\u091b\u093f\u0915 \u091a\u0930, \u091c\u094b \u0917\u0947\u0902\u0926 \u0915\u0940 \u0938\u094d\u0925\u093f\u0924\u093f \u0915\u093e \u092a\u094d\u0930\u0924\u093f\u0928\u093f\u0927\u093f\u0924\u094d\u0935 \u0915\u0930\u0924\u0940 \u0939\u0948, \u0907\u0938\u0915\u093e \u0938\u092e\u093e\u0928 \u0906\u092f\u0924\u093e\u0915\u093e\u0930 \u0935\u093f\u0924\u0930\u0923 \u0939\u094b\u0917\u093e \u0914\u0930 \u0907\u0938\u0947 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim Un(a,b)<\/span>.<\/span> \u0932\u093f\u0916\u093e \u091c\u093e\u090f\u0917\u093e\u0964<\/p>\n<p><a name=\"5\"><\/a><\/br><\/br><\/p>\n<h3>\u0938\u093e\u092e\u093e\u0928\u094d\u092f (\u0917\u0949\u0938\u093f\u092f\u0928) \u0935\u093f\u0924\u0930\u0923<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=1109s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0928\u093f\u0930\u0902\u0924\u0930 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0935\u093f\u0924\u0930\u0923\u094b\u0902 \u092e\u0947\u0902<\/span><\/strong><\/a> \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0935\u093f\u0924\u0930\u0923 \u0935\u094d\u092f\u093e\u0935\u0939\u093e\u0930\u093f\u0915 \u0930\u0942\u092a \u0938\u0947 \u0938\u092c\u0938\u0947 \u0932\u094b\u0915\u092a\u094d\u0930\u093f\u092f \u092e\u0947\u0902 \u0938\u0947 \u090f\u0915 \u0939\u0948\u0964<\/p>\n<h4>\u092e\u093e\u0928\u0915 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0935\u093f\u0924\u0930\u0923<\/h4>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=1150s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u092e\u093e\u0928\u0915 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0918\u0928\u0924\u094d\u0935<\/span><\/strong><\/a> \u0907\u0938 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u0947 \u092e\u093e\u0927\u094d\u092f\u092e \u0938\u0947 \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u093f\u092f\u093e \u0917\u092f\u093e \u0939\u0948:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\phi_{0,1}(x) = \\frac{1}{\\sqrt{2\\pi}} e^{-x^2\/2}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0907\u0938\u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e \u0938\u0947, \u092f\u0939 \u0938\u094d\u092a\u0937\u094d\u091f \u0939\u0948 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi\\gt 0.<\/span><\/span> \u0907\u0938\u0932\u093f\u090f, \u092f\u0939 \u0938\u0924\u094d\u092f\u093e\u092a\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948 \u0915\u093f \u092f\u0939 \u090f\u0915 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0918\u0928\u0924\u094d\u0935 \u0939\u0948 \u092c\u0938 \u092f\u0939 \u0938\u0941\u0928\u093f\u0936\u094d\u091a\u093f\u0924 \u0915\u0930\u0915\u0947 \u0915\u093f<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\int_{-\\infty}^{+\\infty}\\phi_{0,1}(x)dx<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u092f\u0939 \u0905\u0902\u0924\u093f\u092e \u0938\u092e\u093e\u0928\u0924\u093e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I^2<\/span><\/span> \u0915\u0947 \u092e\u0942\u0932\u094d\u092f \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0930\u0915\u0947 \u092a\u094d\u0930\u0926\u0930\u094d\u0936\u093f\u0924 \u0915\u0940 \u091c\u093e \u0938\u0915\u0924\u0940 \u0939\u0948, \u091c\u092c <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I =\\int_{-\\infty}^{+\\infty}\\phi(x)dx=1.<\/span><\/span> \u0935\u093e\u0938\u094d\u0924\u0935 \u092e\u0947\u0902, \u092f\u0939 \u0939\u0948:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\nI^2 &amp; = \\displaystyle \\int_{-\\infty}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}} e^{-x^2\/2} dx \\int_{-\\infty}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}} e^{-x^2\/2}dx \\\\ \\\\\n\n&amp; = \\displaystyle \\int_{-\\infty}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}} e^{-x^2\/2} dx \\int_{-\\infty}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}} e^{-y^2\/2} dy \\\\ \\\\\n\n&amp; = \\displaystyle \\frac{1}{{2\\pi}} \\int_{-\\infty}^{+\\infty} \\int_{-\\infty}^{+\\infty} e^{-\\frac{x^2 + y^2}{2}} dxdy \\\\ \\\\\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0932\u0947\u0915\u093f\u0928 \u092f\u0939 \u0939\u0948 \u0915\u093f<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\displaystyle \\int_{-\\infty}^{+\\infty} \\int_{-\\infty}^{+\\infty} e^{-\\frac{x^2 + y^2}{2}} dxdy = \\int_{0}^{2\\pi} \\int_{0}^{+\\infty} e^{-r^2\/2} rdr d\\theta = 2\\pi <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0907\u0938\u0932\u093f\u090f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I^2 = 1,<\/span><\/span> \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">I=\\int_{-\\infty}^{+\\infty}\\phi_{0,1}(x)dx = 1. <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u092e\u093e\u0928\u0915 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0918\u0928\u0924\u094d\u0935 \u0938\u0947 \u092e\u093e\u0928\u0915 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0935\u093f\u0924\u0930\u0923 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{0,1}(x) = \\int_{-\\infty}^x\\phi_{0,1}(t)dt.<\/span><\/span> \u092f\u0926\u093f \u090f\u0915 \u092f\u093e\u0926\u0943\u091a\u094d\u091b\u093f\u0915 \u091a\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u092e\u0947\u0902 \u092e\u093e\u0928\u0915 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0935\u093f\u0924\u0930\u0923 \u0939\u0948, \u0924\u094b \u0939\u092e \u0932\u093f\u0916\u0924\u0947 \u0939\u0948\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim N(0,1).<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{0,1}(x)<\/span><\/span> \u0935\u093f\u0924\u0930\u0923 \u0915\u094b \u0938\u094d\u092a\u0937\u094d\u091f \u0930\u0942\u092a \u0938\u0947 \u0928\u0939\u0940\u0902 \u0917\u0923\u0928\u093e \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948, \u0932\u0947\u0915\u093f\u0928 \u0910\u0938\u0947 \u091f\u0947\u092c\u0932\u094d\u0938 \u0939\u0948\u0902 \u091c\u094b \u0924\u094d\u0935\u0930\u093f\u0924 \u0930\u0942\u092a \u0938\u0947 \u0905\u0928\u0941\u092e\u093e\u0928\u093f\u0924 \u092e\u093e\u0928 \u092a\u094d\u0930\u0926\u093e\u0928 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902\u0964<\/p>\n<h4>\u092e\u093e\u092a\u0926\u0902\u0921 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma<\/span><\/span> \u0915\u0947 \u0938\u093e\u0925 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0935\u093f\u0924\u0930\u0923<\/h4>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=1875s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u092e\u093e\u0928\u0915 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0935\u093f\u0924\u0930\u0923 \u0918\u0928\u0924\u094d\u0935 \u0938\u0947<\/span><\/strong><\/a> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi_{0,1}<\/span><\/span> \u0938\u0902\u092d\u0935 \u0939\u0948 \u0915\u093f \u092e\u093e\u092a\u0926\u0902\u0921 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma,<\/span><\/span> \u091c\u0939\u093e\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu\\in\\mathbb{R}<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma\\gt 0 <\/span><\/span> \u0939\u0948\u0902, \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0935\u093f\u0924\u0930\u0923 \u0918\u0928\u0924\u094d\u0935 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma<\/span><\/span> \u0915\u0947 \u0938\u093e\u0925 \u092c\u0928\u093e\u0928\u093e \u0938\u0902\u092d\u0935 \u0939\u0948\u0964 \u092f\u0939 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0939\u0948:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\phi_{\\mu,\\sigma}(x) = \\frac{1}{\\sigma}\\phi_{0,1}\\left(\\frac{x-\\mu}{\\sigma} \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0935\u093f\u0924\u0930\u0923 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sigma,<\/span><\/span> \u0915\u0947 \u0938\u093e\u0925 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{\\mu,\\sigma}(x)<\/span><\/span> \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0939\u0948:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\Phi_{\\mu,\\sigma}(x) = \\int_{-\\infty}^x\\frac{1}{\\sigma}\\phi_{0,1}\\left(\\frac{t-\\mu}{\\sigma} \\right)dt = \\frac{1}{\\sqrt{2\\pi\\sigma}}\\int_{-\\infty}^x e^{-\\frac{(t-\\mu)^2}{2\\sigma^2}}dt<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u092f\u0926\u093f \u090f\u0915 \u092f\u093e\u0926\u0943\u091a\u094d\u091b\u093f\u0915 \u091a\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u092e\u0947\u0902 \u0938\u093e\u092e\u093e\u0928\u094d\u092f \u0935\u093f\u0924\u0930\u0923 \u092e\u093e\u092a\u0926\u0902\u0921 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu, \\sigma,<\/span><\/span> \u0915\u0947 \u0938\u093e\u0925 \u0939\u0948, \u0924\u094b \u0939\u092e \u0932\u093f\u0916\u0924\u0947 \u0939\u0948\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim N(\\mu, \\sigma).<\/span><\/span><\/p>\n<p><a name=\"6\"><\/a><\/br><\/br><\/p>\n<h3>\u0935\u0947\u0907\u092c\u0941\u0932 \u0935\u093f\u0924\u0930\u0923<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=2230s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0935\u0947\u0907\u092c\u0941\u0932 \u0935\u093f\u0924\u0930\u0923<\/span><\/strong><\/a> \u092e\u093e\u092a\u0926\u0902\u0921 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha,\\beta \\gt 0<\/span><\/span> \u0915\u0947 \u0938\u093e\u0925 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0940 \u090f\u0915 \u0935\u093f\u0924\u0930\u0923 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0939\u0948:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(t) = \\left\\{\\begin{array}{llr}\n\n\\left(1 - e^{-t\/\\alpha} \\right)^\\beta &amp;;&amp; t\\geq 0 \\\\ \\\\\n\n0 &amp;;&amp; t\\lt 0\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u092f\u0926\u093f \u090f\u0915 \u092f\u093e\u0926\u0943\u091a\u094d\u091b\u093f\u0915 \u091a\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u092e\u0947\u0902 \u092e\u093e\u092a\u0926\u0902\u0921 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta<\/span><\/span> \u0915\u0947 \u0938\u093e\u0925 \u0935\u0947\u0907\u092c\u0941\u0932 \u0935\u093f\u0924\u0930\u0923 \u0939\u0948, \u0924\u094b \u0939\u092e \u0932\u093f\u0916\u0924\u0947 \u0939\u0948\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim We(\\alpha,\\beta).<\/span><\/span> \u0935\u0947\u0907\u092c\u0941\u0932 \u0935\u093f\u0924\u0930\u0923 \u090f\u0915\u094d\u0938\u092a\u094b\u0928\u0947\u0936\u093f\u092f\u0932 \u0935\u093f\u0924\u0930\u0923 \u0915\u093e \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0940\u0915\u0930\u0923 \u0939\u0948, \u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0902 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">We(\\alpha,1) = Ex(\\alpha).<\/span><\/span><\/p>\n<p><a name=\"7\"><\/a><\/br><\/br><\/p>\n<h3>\u0917\u093e\u092e\u093e \u0935\u093f\u0924\u0930\u0923<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=REOTUa7K8uQ&amp;t=2311s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u0917\u093e\u092e\u093e \u0935\u093f\u0924\u0930\u0923<\/span><\/strong><\/a> \u092e\u093e\u092a\u0926\u0902\u0921 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta,\\alpha<\/span><\/span> \u0915\u0947 \u0938\u093e\u0925 \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u0940 \u0918\u0928\u0924\u094d\u0935 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0939\u0948:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f(t) = \\left\\{\\begin{array}{llr}\n\n\\displaystyle \\frac{1}{\\alpha \\Gamma(\\beta)}\\left(\\frac{t}{\\alpha} \\right)^{\\beta-1}e^{-t\/\\alpha} &amp;;&amp; t\\geq 0 \\\\ \\\\\n\n0 &amp;;&amp; t\\lt 0\n\n\\end{array}\\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u091c\u0939\u093e\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma(s) = \\displaystyle \\int_0^{+\\infty}u^{s-1}e^{-u}du <\/span><\/span> \u0935\u0939 \u0939\u0948 \u091c\u093f\u0938\u0947 \u00ab\u0917\u093e\u092e\u093e \u092b\u093c\u0902\u0915\u094d\u0936\u0928\u00bb \u0915\u0939\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964<\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0917\u093e\u092e\u093e \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u0940 \u0938\u092c\u0938\u0947 \u0909\u0932\u094d\u0932\u0947\u0916\u0928\u0940\u092f \u0935\u093f\u0936\u0947\u0937\u0924\u093e\u0913\u0902 \u092e\u0947\u0902 \u0938\u0947 \u090f\u0915 \u092f\u0939 \u0939\u0948 \u0915\u093f \u092f\u0939 \u092a\u094d\u0930\u093e\u0915\u0943\u0924\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u092b\u0948\u0915\u094d\u091f\u0930\u093f\u092f\u0932\u094d\u0938 \u0915\u094b \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 (\u092f\u0939\u093e\u0902 \u0924\u0915 \u0915\u093f \u091c\u091f\u093f\u0932 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902) \u092a\u0930 \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0940\u0915\u0943\u0924 \u0915\u0930\u0928\u0947 \u0915\u0940 \u0905\u0928\u0941\u092e\u0924\u093f \u0926\u0947\u0924\u093e \u0939\u0948\u0964 \u092f\u0939 \u091c\u093e\u0901\u091a\u0928\u093e \u0915\u0920\u093f\u0928 \u0928\u0939\u0940\u0902 \u0939\u0948 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma(s+1) = s\\Gamma(s)<\/span><\/span> \u092d\u093e\u0917\u094b\u0902 \u0915\u0947 \u090f\u0915\u0940\u0915\u0930\u0923 \u0938\u0947\u0964 \u0907\u0938\u0915\u0947 \u0905\u0932\u093e\u0935\u093e, \u091c\u0948\u0938\u093e \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Gamma(1)=1<\/span><\/span> \u0939\u0948:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\forall n\\in\\mathbb{N}\\right)\\left(\\Gamma(n) = (n-1)! \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u092f\u0926\u093f \u090f\u0915 \u092f\u093e\u0926\u0943\u091a\u094d\u091b\u093f\u0915 \u091a\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> \u092e\u0947\u0902 \u092e\u093e\u092a\u0926\u0902\u0921 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta, \\alpha<\/span><\/span> \u0915\u0947 \u0938\u093e\u0925 \u0917\u093e\u092e\u093e \u0935\u093f\u0924\u0930\u0923 \u0939\u0948, \u0924\u094b \u0939\u092e \u0932\u093f\u0916\u0924\u0947 \u0939\u0948\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim Ga(\\alpha,\\beta).<\/span><\/span> \u0917\u093e\u092e\u093e \u0935\u093f\u0924\u0930\u0923 \u090f\u0915\u094d\u0938\u092a\u094b\u0928\u0947\u0936\u093f\u092f\u0932 \u0935\u093f\u0924\u0930\u0923 \u0915\u093e \u090f\u0915 \u0905\u0928\u094d\u092f \u0938\u093e\u092e\u093e\u0928\u094d\u092f\u0940\u0915\u0930\u0923 \u0939\u0948, \u0927\u094d\u092f\u093e\u0928 \u0926\u0947\u0902 \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Ga(\\alpha,1) = Ex(\\alpha).<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u092a\u0949\u0907\u0938\u0928 \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e \u092e\u0947\u0902 \u0906\u0935\u0943\u0924\u094d\u0924\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> \u0915\u0947 \u0938\u093e\u0925 (\u091c\u0948\u0938\u0947 \u0930\u0947\u0921\u093f\u092f\u094b\u0927\u0930\u094d\u092e\u0940 \u0915\u094d\u0937\u092f), \u092f\u0926\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T<\/span><\/span> \u0935\u0939 \u092f\u093e\u0926\u0943\u091a\u094d\u091b\u093f\u0915 \u091a\u0930 \u0939\u0948 \u091c\u094b \u092e-\u0935\u0947\u0902 \u0918\u091f\u0928\u093e \u0915\u0947 \u0939\u094b\u0928\u0947 \u0915\u0947 \u0938\u092e\u092f \u0915\u093e \u092a\u094d\u0930\u0924\u093f\u0928\u093f\u0927\u093f\u0924\u094d\u0935 \u0915\u0930\u0924\u093e \u0939\u0948; \u0924\u092c, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t\\geq 0<\/span><\/span> \u0914\u0930 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">N<\/span><\/span> \u0918\u091f\u0928\u093e\u0913\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u0947 \u0932\u093f\u090f \u091c\u094b \u0938\u092e\u092f \u0905\u0902\u0924\u0930\u093e\u0932 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">[0,t]<\/span><\/span> \u092e\u0947\u0902 \u0939\u094b\u0924\u0940 \u0939\u0948\u0902, \u092f\u0939 \u0939\u094b\u0917\u093e \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t\\lt T \\leftrightarrow N\\lt m<\/span><\/span> \u0914\u0930, \u091c\u0948\u0938\u093e \u0915\u093f <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">N\\sim Po(ct),<\/span><\/span> \u092f\u0939 \u0939\u0948:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1-F_T(t) = P(T\\gt t) = \\displaystyle \\sum_{k=0}^{m-1}Po(k; ct)=e^{-ct}\\sum_{k=0}^{m-1}\\frac{(ct)^k}{k!}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0914\u0930 \u0907\u0938\u0932\u093f\u090f, \u092f\u0926\u093f \u0939\u092e \u0907\u0938\u0947 \u0935\u094d\u092f\u0941\u0924\u094d\u092a\u0928\u094d\u0928 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u0939\u092e\u0947\u0902 \u092e\u093f\u0932\u0947\u0917\u093e \u0915\u093f \u0918\u0928\u0924\u094d\u0935 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0939\u0948<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f(t) = ce^{-ct}\\frac{(ct)^{m-1}}{(m-1)!}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0914\u0930 \u0907\u0938\u0932\u093f\u090f, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T\\sim Ga(1\/c, m).<\/span><\/span><\/p>\n<p><a name=\"8\"><\/a><\/br><\/br><\/p>\n<h2>\u0905\u092d\u094d\u092f\u093e\u0938<\/h2>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>\u0927\u0928\u0924\u094d\u0935 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u0947 \u0932\u093f\u090f \u0928\u093f\u0930\u0902\u0924\u0930\u0924\u093e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> \u0915\u093e \u092e\u093e\u0928 \u0928\u093f\u0915\u093e\u0932\u0947\u0902 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle f(x) = \\frac{c}{x^2+1}<\/span><\/span> \u0914\u0930 \u0907\u0938\u0915\u0947 \u0935\u093f\u0924\u0930\u0923 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0930\u0947\u0902 (\u0915\u094c\u0936\u0940 \u0935\u093f\u0924\u0930\u0923)<\/li>\n<li>\u0938\u092e\u093e\u0928 \u0935\u093f\u0924\u0930\u0923 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Un(a.b),<\/span><\/span> \u0915\u0940 \u0918\u0928\u0924\u094d\u0935 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0938\u0947 \u0907\u0938\u0915\u093e \u0935\u093f\u0924\u0930\u0923 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0928\u093f\u0915\u093e\u0932\u0947\u0902\u0964<\/li>\n<li>\u0938\u093e\u092c\u093f\u0924 \u0915\u0930\u0947\u0902 \u0915\u093f \u092b\u093c\u0902\u0915\u094d\u0936\u0928 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{\\mu,\\sigma}(x)<\/span><\/span> \u090f\u0915 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0935\u093f\u0924\u0930\u0923 \u092b\u093c\u0902\u0915\u094d\u0936\u0928 \u0939\u0948\u0964<\/li>\n<\/ol>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/kdxgrB1h98g\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0928\u093f\u0930\u0902\u0924\u0930 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0935\u093f\u0924\u0930\u0923 \u0938\u093e\u0930\u093e\u0902\u0936\u092f\u0939\u093e\u0902 \u0939\u092e \u0928\u093f\u0930\u0902\u0924\u0930 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0935\u093f\u0924\u0930\u0923 \u0915\u0940 \u0905\u0935\u0927\u093e\u0930\u0923\u093e \u0915\u0940 \u0917\u0939\u0930\u093e\u0908 \u0938\u0947 \u091c\u093e\u0902\u091a \u0915\u0930\u0947\u0902\u0917\u0947, \u092a\u093e\u0902\u091a \u0938\u092c\u0938\u0947 \u092a\u094d\u0930\u0938\u093f\u0926\u094d\u0927 \u0935\u093f\u0924\u0930\u0923\u094b\u0902 \u0915\u0940 \u0935\u093f\u0936\u0947\u0937\u0924\u093e\u0913\u0902 \u0914\u0930 \u0909\u092a\u092f\u094b\u0917\u094b\u0902 \u092a\u0930 \u092a\u094d\u0930\u0915\u093e\u0936 \u0921\u093e\u0932\u0924\u0947 \u0939\u0941\u090f: \u090f\u0915\u094d\u0938\u092a\u094b\u0928\u0947\u0936\u093f\u092f\u0932 \u0935\u093f\u0924\u0930\u0923, \u0938\u092e\u093e\u0928 \u0906\u092f\u0924\u093e\u0915\u093e\u0930 \u0935\u093f\u0924\u0930\u0923, \u0938\u093e\u092e\u093e\u0928\u094d\u092f (\u0917\u0949\u0938\u093f\u092f\u0928) \u0935\u093f\u0924\u0930\u0923, \u0935\u0947\u0907\u092c\u0941\u0932 \u0935\u093f\u0924\u0930\u0923 \u0914\u0930 \u0917\u093e\u092e\u093e \u0935\u093f\u0924\u0930\u0923\u0964 \u0907\u0928 \u0935\u093f\u0924\u0930\u0923\u094b\u0902 \u092e\u0947\u0902 \u0938\u0947 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0915\u094b \u092a\u0930\u093f\u092d\u093e\u0937\u093f\u0924 \u0915\u0930\u0928\u0947 \u0935\u093e\u0932\u0947 \u0917\u0923\u093f\u0924\u0940\u092f \u0938\u0942\u0924\u094d\u0930 \u092a\u094d\u0930\u0926\u093e\u0928 \u0915\u093f\u090f \u0917\u090f \u0939\u0948\u0902, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":26864,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":0,"footnotes":""},"categories":[577,678],"tags":[],"class_list":["post-27277","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-577","category-678"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u0928\u093f\u0930\u0902\u0924\u0930 \u0938\u0902\u092d\u093e\u0935\u094d\u092f\u0924\u093e \u0935\u093f\u0924\u0930\u0923 - 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