{"id":10482,"date":"2020-10-30T15:26:42","date_gmt":"2020-10-30T14:26:42","guid":{"rendered":"https:\/\/complex-systems-ai.com\/?page_id=10482"},"modified":"2024-02-13T08:04:15","modified_gmt":"2024-02-13T07:04:15","slug":"exo-chaines-de-markov","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/markov-process\/discrete-time-markov-chain-corrective-exercises\/","title":{"rendered":"9 Corrected exercises: Markov chains in discrete time"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"10482\" class=\"elementor elementor-10482\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8c9b3a5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8c9b3a5\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-845cc92\" data-id=\"845cc92\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-51f9d1f elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"51f9d1f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Stochastic process<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-4008ffd\" data-id=\"4008ffd\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a2c1821 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"a2c1821\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/en\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span 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rel=\"noopener\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Wiki<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6e6b5cb elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6e6b5cb\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e87f64f\" data-id=\"e87f64f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-50f6577 elementor-widget elementor-widget-heading\" data-id=\"50f6577\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewbox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewbox=\"0 0 24 24\" version=\"1.2\" baseprofile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/discrete-time-markov-chain-corrective-exercises\/#Exercices-corriges-les-chaines-de-Markov-en-temps-discret\" >Corrected exercises: Markov chains in discrete time<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/discrete-time-markov-chain-corrective-exercises\/#Exercice-1\" >Exercise 1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/discrete-time-markov-chain-corrective-exercises\/#Exercice-2\" >Exercise 2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/discrete-time-markov-chain-corrective-exercises\/#Exercice-3\" >Exercise 3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/discrete-time-markov-chain-corrective-exercises\/#Exercice-4\" >Exercise 4<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/discrete-time-markov-chain-corrective-exercises\/#Exercice-5\" >Exercise 5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/discrete-time-markov-chain-corrective-exercises\/#Exercice-6\" >Exercise 6<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/discrete-time-markov-chain-corrective-exercises\/#Exercice-7\" >Exercise 7<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/discrete-time-markov-chain-corrective-exercises\/#Exercice-8\" >Exercise 8<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/discrete-time-markov-chain-corrective-exercises\/#Exercice-9\" >Exercise 9<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercices-corriges-les-chaines-de-Markov-en-temps-discret\"><\/span>Corrected exercises: Markov chains in discrete time<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c212e6a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c212e6a\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f623a06\" data-id=\"f623a06\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f868867 elementor-widget elementor-widget-text-editor\" data-id=\"f868867\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>This page includes many corrected exercises on Markov chains in discrete time and asymptotic behavior, class, chain <a href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/invariant-law-and-asymptotic-behavior\/\">ergodic<\/a>, uptake.<\/p><p><img decoding=\"async\" class=\"aligncenter wp-image-11096 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/09\/cropped-Capture.png\" alt=\"discrete time Markov chains\" width=\"97\" height=\"97\" title=\"\"><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-53047c6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"53047c6\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f41f63d\" data-id=\"f41f63d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-855199d elementor-widget elementor-widget-heading\" data-id=\"855199d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-1\"><\/span>Exercise 1<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1eb1cfe elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1eb1cfe\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-00a396e\" data-id=\"00a396e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fac9ef7 elementor-widget elementor-widget-text-editor\" data-id=\"fac9ef7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Build the <a href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/discrete-time-markov-chains\/\">markov chain<\/a> corresponding to the following stochastic matrix:<\/p><table><tbody><tr><td width=\"204\"><p>0.4<\/p><\/td><td width=\"204\"><p>0.6<\/p><\/td><td width=\"204\"><p>0<\/p><\/td><\/tr><tr><td width=\"204\"><p>0.2<\/p><\/td><td width=\"204\"><p>0.5<\/p><\/td><td width=\"204\"><p>0.3<\/p><\/td><\/tr><tr><td width=\"204\"><p>0<\/p><\/td><td width=\"204\"><p>0.4<\/p><\/td><td width=\"204\"><p>0.6<\/p><\/td><\/tr><\/tbody><\/table><p>How much class does the chain have? Is it reducible? If so, reduce it. Calculate the stationary probability, if it exists, of the irreducible chain. Calculate the periodicity of the classes.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-355cd74 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"355cd74\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d9e92f4\" data-id=\"d9e92f4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-44fb44f elementor-widget elementor-widget-toggle\" data-id=\"44fb44f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-7231\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-7231\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-7231\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-7231\"><p>To represent the chain, we choose to number the states from 1 to 3, in the order of the rows of the transition matrix:<\/p><p><img fetchpriority=\"high\" decoding=\"async\" class=\"aligncenter wp-image-10488 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image67.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"768\" height=\"310\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image67.png 768w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image67-300x121.png 300w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/><\/p><p>To check the classes in the chain, we need to check the communicating states. Here, 1 communicates with 2 and 2 with 3 (therefore accessible in both directions), by transitivity 1 communicates with 3. The chain has only one class, it is irreducible. The states are therefore all recurrent, and are aperiodic because they all have a loop on themselves. Under these conditions, the chain admits a unique stationary probability. It suffices to solve the following system of equations:<\/p><p><img decoding=\"async\" class=\"aligncenter wp-image-10489 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image68.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"379\" height=\"140\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image68.png 379w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image68-300x111.png 300w\" sizes=\"(max-width: 379px) 100vw, 379px\" \/><\/p><p>This gives the vector (4\/25, 12\/25, 9\/25).<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6db31bd elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6db31bd\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d1260f8\" data-id=\"d1260f8\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1486cc2 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"1486cc2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-12fb467 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"12fb467\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-51079c4\" data-id=\"51079c4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0a8b835 elementor-widget elementor-widget-heading\" data-id=\"0a8b835\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-2\"><\/span>Exercise 2<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b7c5803 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b7c5803\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-b7f292d\" data-id=\"b7f292d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-58632cd elementor-widget elementor-widget-text-editor\" data-id=\"58632cd\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>As for Exercise 1 with the following stochastic matrix:<\/p><table><tbody><tr><td width=\"153\"><p>0<\/p><\/td><td width=\"153\"><p>0.5<\/p><\/td><td width=\"153\"><p>0.5<\/p><\/td><td width=\"153\"><p>0<\/p><\/td><\/tr><tr><td width=\"153\"><p>0<\/p><\/td><td width=\"153\"><p>1<\/p><\/td><td width=\"153\"><p>0<\/p><\/td><td width=\"153\"><p>0<\/p><\/td><\/tr><tr><td width=\"153\"><p>0<\/p><\/td><td width=\"153\"><p>0<\/p><\/td><td width=\"153\"><p>0<\/p><\/td><td width=\"153\"><p>1<\/p><\/td><\/tr><tr><td width=\"153\"><p>0<\/p><\/td><td width=\"153\"><p>0<\/p><\/td><td width=\"153\"><p>1<\/p><\/td><td width=\"153\"><p>0<\/p><\/td><\/tr><\/tbody><\/table>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-12e63a7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"12e63a7\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-97c7436\" data-id=\"97c7436\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c90c01a elementor-widget elementor-widget-toggle\" data-id=\"c90c01a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-2101\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-2101\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-2101\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-2101\"><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10490 size-medium\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image69-300x177.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"300\" height=\"177\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image69-300x177.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image69.png 579w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p><p>We note that 2 does not communicate with anyone, it is an absorbing state. Nobody communicates with 1, the state forms a <a href=\"https:\/\/complex-systems-ai.com\/en\/markov-process\/recurrence-and-transition-criteria\/\">transitional class<\/a>. States 3 and 4 communicate with each other, they form a recurrent class. So there are three classes {1}, {2}, {3,4}.<\/p><p>There is no guarantee of stationary probability. Let&#039;s solve the following system:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10491 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image70.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"426\" height=\"160\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image70.png 426w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image70-300x113.png 300w\" sizes=\"(max-width: 426px) 100vw, 426px\" \/><\/p><p>The system does not allow a single solution. Let Pi2 be between 0 and 1, then we find a solution:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10492 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image71.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"166\" height=\"140\" title=\"\"><\/p><p>We notice in the class {3,4} that the probabilities are equal on a frequency of 2k, we deduce that the class is periodic over a period of 2. The class {2} is absorbing, there is no period. The class {1} is transient, which is why there is no longer a population after a time k.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-cb7b427 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"cb7b427\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2717ac2\" data-id=\"2717ac2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3ba68ea elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"3ba68ea\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-bed471e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"bed471e\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9ad4cca\" data-id=\"9ad4cca\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c8f0ff9 elementor-widget elementor-widget-heading\" data-id=\"c8f0ff9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-3\"><\/span>Exercise 3<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ab5ab6b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ab5ab6b\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3506c04\" data-id=\"3506c04\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e23d574 elementor-widget elementor-widget-text-editor\" data-id=\"e23d574\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>As for Exercise 1 with the following stochastic matrix:<\/p><table><tbody><tr><td width=\"204\"><p>0<\/p><\/td><td width=\"204\"><p>1<\/p><\/td><td width=\"204\"><p>0<\/p><\/td><\/tr><tr><td width=\"204\"><p>0<\/p><\/td><td width=\"204\"><p>0<\/p><\/td><td width=\"204\"><p>1<\/p><\/td><\/tr><tr><td width=\"204\"><p>1<\/p><\/td><td width=\"204\"><p>0<\/p><\/td><td width=\"204\"><p>0<\/p><\/td><\/tr><\/tbody><\/table>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d3f64d2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d3f64d2\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d6afe98\" data-id=\"d6afe98\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-994b1cf elementor-widget elementor-widget-toggle\" data-id=\"994b1cf\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1601\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1601\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1601\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1601\"><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10493 size-medium\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image72-300x236.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"300\" height=\"236\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image72-300x236.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image72.png 340w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p><p>Now let&#039;s move on to the stationary probability:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10495 size-medium\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image73-300x119.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"300\" height=\"119\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image73-300x119.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image73.png 326w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p><p>Which gives the vector (1\/3, 1\/3, 1\/3). Regarding the periodicity, we notice that the transition matrix becomes the identity matrix when it is cubed. We can therefore deduce that the Markov chain is periodic with periodicity 3. The stationary probability therefore does not exist, on the other hand we can conclude that over a large random step, we spend 1\/3 of the time in each of the states.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ad6f7d0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ad6f7d0\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e0eff04\" data-id=\"e0eff04\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-97c5cd4 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"97c5cd4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5607906 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5607906\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4d2b09d\" data-id=\"4d2b09d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fa7a503 elementor-widget elementor-widget-heading\" data-id=\"fa7a503\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-4\"><\/span>Exercise 4<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-396ed62 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"396ed62\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-7eafc81\" data-id=\"7eafc81\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2fa2798 elementor-widget elementor-widget-text-editor\" data-id=\"2fa2798\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>From the following three transition graphs, reconstruct the associated Markov chains (state space and matrix). Then do the analysis of these Markov chains: the classes and characteristics, stationary law, absorption probability, mean absorption times).<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10496 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image75.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"394\" height=\"240\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image75.png 394w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image75-300x183.png 300w\" sizes=\"(max-width: 394px) 100vw, 394px\" \/><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e79ce1b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e79ce1b\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-966141c\" data-id=\"966141c\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-030cd6f elementor-widget elementor-widget-toggle\" data-id=\"030cd6f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-3191\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-3191\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-3191\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-3191\"><p>FIRST <a href=\"https:\/\/complex-systems-ai.com\/en\/graph-theory-2\/\">CHART<\/a>\u00a0:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10497 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image76.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"490\" height=\"127\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image76.png 490w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image76-300x78.png 300w\" sizes=\"(max-width: 490px) 100vw, 490px\" \/><\/p><p>All the states communicate, so there is only one necessarily recurring class. Note that the power matrix 4 gives the identity matrix, the periodicity is therefore 4.<\/p><p>Now let&#039;s calculate the stationary distribution:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10498 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image77.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"648\" height=\"151\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image77.png 648w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image77-300x70.png 300w\" sizes=\"(max-width: 648px) 100vw, 648px\" \/><\/p><p>Since there is periodicity, we know that there is no convergence. In addition, a random step will spend \u00bc of its time in each state. Regarding absorption, we have an irreducible chain, so the probability of being in the unique class is 1.<\/p><p>SECOND GRAPH:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10499 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image78.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"465\" height=\"117\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image78.png 465w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image78-300x75.png 300w\" sizes=\"(max-width: 465px) 100vw, 465px\" \/><\/p><p>All the states communicate, the states are all of the same period. The matrix seems similar to the first graph, but note that state 1 has a loop, therefore of period 1. The unique class is therefore aperiodic (of period 1).<\/p><p>As the chain is aperiodic and irreducible, we know that it converges towards the stationary distribution.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10500 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image79.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"655\" height=\"156\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image79.png 655w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image79-300x71.png 300w\" sizes=\"(max-width: 655px) 100vw, 655px\" \/><\/p><p>Since there is only one class, the calculation of absorption probabilities is simple, from any state, we always go in the same class.<\/p><p>THIRD GRAPH:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10507 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image80.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"474\" height=\"144\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image80.png 474w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image80-300x91.png 300w\" sizes=\"(max-width: 474px) 100vw, 474px\" \/><\/p><p>We have four classes<\/p><p>{1} which is transitory therefore no periodicity.<\/p><p>[2} which is recurrent with a loop therefore a periodicity of 1.<\/p><p>{3} same as {1}<\/p><p>{4} same as {2}<\/p><p>Let us calculate the stationary distribution. Given that there are transient classes, we know that there will be no convergence:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10508 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image81.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"641\" height=\"136\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image81.png 641w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image81-300x64.png 300w\" sizes=\"(max-width: 641px) 100vw, 641px\" \/><\/p><p>Note that there is an infinite number of solutions.<\/p><p>Let us calculate the absorption probabilities. Let\u2019s start with state 2. Let\u2019s calculate the vector v such that v<sub>i<\/sub>\u00a0is equal to the probability that the chain is absorbed in 2 knowing that it starts at i. The calculations are simple since we have two transient classes of expectation 1 (loop-free and single-state).<\/p><p>We therefore know that state 2 is reached with a step of 1 maximum, this considerably reduces the number of calculations. Indeed, the probability of reaching 2 starting from 1 is known since it is p<sub>12<\/sub>. The transition probabilities are therefore the second column of P. Similarly the absorption probabilities of state 4 is the fourth column of P. In the general case, it is necessary to calculate the vector v such that (PI) v = 0 .<\/p><p>We can calculate the average absorption time. States 2 and 4 are recurrent therefore x<sub>2<\/sub>= x<sub>4<\/sub>= 0. It is quick to see that x<sub>1<\/sub>= x<sub>3<\/sub>=1+0=1.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-37f6804 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"37f6804\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-588a210\" data-id=\"588a210\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-acf9c15 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"acf9c15\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-31ed474 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"31ed474\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-7c456ee\" data-id=\"7c456ee\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f498689 elementor-widget elementor-widget-heading\" data-id=\"f498689\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-5\"><\/span>Exercise 5<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f0285b5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f0285b5\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c092762\" data-id=\"c092762\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2a847e1 elementor-widget elementor-widget-text-editor\" data-id=\"2a847e1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Consider the following example:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10509 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image82.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"334\" height=\"119\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image82.png 334w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image82-300x107.png 300w\" sizes=\"(max-width: 334px) 100vw, 334px\" \/><\/p><p>Calculate the probability of the following trajectories (h, a, f, h), (h, a, f, a), (a, a, a).<\/p><p>Calculate the distribution of states X1 at time t = 1 if we assume X0 = (1, 0, 0).<\/p><p>To interpret.<\/p><p>Show that a uniform distribution X0 = (1\/3, 1\/3, 1\/3) is not a stationary distribution for this Markov chain.<\/p><p>Is there a stationary distribution for this Markov chain?<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9b0fd59 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9b0fd59\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-261f2df\" data-id=\"261f2df\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-79fe9b5 elementor-widget elementor-widget-toggle\" data-id=\"79fe9b5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1271\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1271\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1271\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1271\"><p>P (h, a, f, h) = X0 (h) * 0.45 * 0.4 * 0 = 0<\/p><p>P (h, a, f, a) = 0.018 * X0 (h)<\/p><p>P (a, a, a) = 0.25 * X0 (a)<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10510 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image83.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"611\" height=\"94\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image83.png 611w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image83-300x46.png 300w\" sizes=\"(max-width: 611px) 100vw, 611px\" \/><\/p><p>So there is, after three iterations, half of the plot covered with grass, .45 shrubs and .05 with forest assuming that at first there is only grass.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10511 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image84.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"601\" height=\"87\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image84.png 601w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image84-300x43.png 300w\" sizes=\"(max-width: 601px) 100vw, 601px\" \/><\/p><p>The uniform distribution is not stationary. To calculate the stationary distribution it is necessary to solve the system XP = X:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10512 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image85.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"348\" height=\"93\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image85.png 348w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image85-300x80.png 300w\" sizes=\"(max-width: 348px) 100vw, 348px\" \/><\/p><p>Which gives the solution (2\/53, 10\/53, 41\/53). We can conclude that we tend towards a majority of forest, with a density of 41\/53.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a5c5859 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a5c5859\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2ac8182\" data-id=\"2ac8182\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1e00d26 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"1e00d26\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7f287a6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7f287a6\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9288637\" data-id=\"9288637\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-beb4e8e elementor-widget elementor-widget-heading\" data-id=\"beb4e8e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-6\"><\/span>Exercise 6<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7f69242 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7f69242\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-44f47d4\" data-id=\"44f47d4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2765717 elementor-widget elementor-widget-text-editor\" data-id=\"2765717\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>We consider a Markov chain of four states according to the following transition matrix:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10513 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image86.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"269\" height=\"101\" title=\"\"><\/p><p>Determine the classes of the chain then the probability of absorption of state 4 starting from 2.<\/p><p>Determine the absorption time in 1 or 4 from 2.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d98081f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d98081f\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4f03bc9\" data-id=\"4f03bc9\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-798ccda elementor-widget elementor-widget-toggle\" data-id=\"798ccda\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1271\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1271\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1271\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1271\"><p>We notice that state 1 and state 4 are both absorbing states, forming two classes. States 2 and 3 communicate but can go in an absorbing class. We deduce that {2,3} forms a transient class.<\/p><p>According to the course, the absorption probability of an absorbing state k from a state i is solved by the following system:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10514 size-medium\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image87-300x95.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"300\" height=\"95\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image87-300x95.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image87.png 314w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p><p>This gives the following system:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10515 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image87-1.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"193\" height=\"100\" title=\"\"><\/p><p>And so the solution vector (0; 1\/3; 2\/3; 1), h<sub>1<\/sub>= 0 so as not to influence the other equations. We deduce by h<sub>2<\/sub>\u00a0the probability of absorption of state 4 starting from 2.<\/p><p>According to the course, the absorption time by an absorbing state k from a state i is solved by the following system:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10516 size-medium\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Capture-300x78.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"300\" height=\"78\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Capture-300x78.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Capture.png 329w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p><p>This gives the following system:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10517 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image88.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"173\" height=\"96\" title=\"\"><\/p><p>And so the solution vector is (0; 2; 2; 0). The absorption time K<sub>2<\/sub>\u00a0give the correct answer.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a20ff26 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a20ff26\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2cf8c34\" data-id=\"2cf8c34\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-54392b0 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"54392b0\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e1f9e22 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e1f9e22\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ca7afbc\" data-id=\"ca7afbc\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-cafbfad elementor-widget elementor-widget-heading\" data-id=\"cafbfad\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-7\"><\/span>Exercise 7<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f06991f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f06991f\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-221e8d0\" data-id=\"221e8d0\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2ff6137 elementor-widget elementor-widget-text-editor\" data-id=\"2ff6137\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>We consider a road network made up of 5 cities A, B, C, D, S as follows:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10518 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image89.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"240\" height=\"222\" title=\"\"><\/p><p>To model the road traffic, we consider that the cars take a road leaving the city in an equitable (uniform) way. Denote by X<sub>not<\/sub>\u00a0the random variable representing the city reached by a car at time n.<\/p><ol><li>Justify that the probabilistic process is indeed stochastic and forms a Markov chain. And give the transition matrix.<\/li><li>What is the probability that, starting from B, the car will be in C after two iterations.<\/li><li>Classify states into classes and calculate their periods.<\/li><li>Write the system of equations verified by stationary probabilities.<\/li><li>We denote the number of routes resulting from i. Deduce a distribution of an initial population as a function of. Calculate the stationary distribution. Is she unique?<\/li><li>A car is observed for a time N which is assumed to be very large. Approximately how long has she spent in a given state?<\/li><li>Starting from S, what is the average number of steps it takes for the car to come back to S?<\/li><li>What is the probability that, starting from S, the car visits city D before city C?<\/li><\/ol>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-af6c9e2 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"af6c9e2\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e0a5428\" data-id=\"e0a5428\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1df4ef9 elementor-widget elementor-widget-toggle\" data-id=\"1df4ef9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-3141\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-3141\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-3141\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-3141\"><ol><li>Even knowing its entire past trajectory, the car&#039;s destination is only decided from its current position. In addition, according to the definition of the exercise, the car has a probability of 1 of leaving its current city, and therefore of taking a step towards connected cities. The choice of step is independent of the past, we are indeed in the presence of a Markov chain. The transition matrix is as follows: <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10519 size-medium\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image90-300x127.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"300\" height=\"127\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image90-300x127.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image90.png 379w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/li><li>There are only two paths that go from B to C in two steps: BAC and BSC. The probability of going from B to C in two steps is therefore equal to \u00bd * 1\/3 +1\/2 * \u00bc = 7\/24.<\/li><li>As the graph shows, all the states are linked to each other, they all communicate and therefore only form a single recurring class (therefore positive). The chain is irreducible. Regarding the periodicity, note that there are chains of all lengths from A to A. The chain is therefore aperiodic.<\/li><li>See the course.<\/li><li>Note that the distribution vector calculated by the number of exits from each city corresponds to the vector of stationary probabilities (1\/4, 1\/6, 1\/6, 1\/12, 1\/3). This probability is unique since the chain is ergodic.<\/li><li>As the chain is ergodic, we know that its asymptotic behavior is equivalent to the stationary probabilities. For example, the car will spend 1\/12 of its time in state D.<\/li><li>Once again, the chain is ergodic, we know that the expectation is the inverse of the stationary probability. The average time to return to S is therefore 3 steps.<\/li><li>This point is much more complex to deduce. The idea is to make the Markov chain absorbent in C or D. The probability, starting from S, of reaching D is therefore the probability of being absorbed by the class {D}. First, we modify states C and D as follows:\u00a0 <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10520 size-medium\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image91-300x114.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"300\" height=\"114\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image91-300x114.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image91.png 381w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><p>We then have two recurrent (and absorbing) classes and a transition class containing {A, B, S}. It only remains to calculate the absorption probabilities and conclude (see the course).<\/p><\/li><\/ol><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-269456c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"269456c\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1296597\" data-id=\"1296597\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0b392c2 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"0b392c2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c3ff8ea elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c3ff8ea\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-90d73d9\" data-id=\"90d73d9\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-cce60b2 elementor-widget elementor-widget-heading\" data-id=\"cce60b2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-8\"><\/span>Exercise 8<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9510594 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9510594\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-cd314ca\" data-id=\"cd314ca\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5542f14 elementor-widget elementor-widget-text-editor\" data-id=\"5542f14\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>We consider a Markov chain with three states with the following transition matrix:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10521 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image92.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"224\" height=\"80\" title=\"\"><\/p><p>What are the recurrent, transient, steady state states? Calculate the time to reach each state. Calculate the period of each state. Deduce on the stationary distribution.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-37a44e5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"37a44e5\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-456e9e6\" data-id=\"456e9e6\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-58addce elementor-widget elementor-widget-toggle\" data-id=\"58addce\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-9291\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-9291\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-9291\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-9291\"><p>From state 1 we can go to state 2 and 3, from state 2 we can go to state 1 and 3, from state 3 we can go to 1 and by transitivity from 1 to 2 we deduce that 3 can go to 2. All the states communicate therefore the chain is irreducible, all the states are positive recurrent.<\/p><p>The chain therefore admits a unique stationary law of value (4\/9; 2\/9; 3\/9)<\/p><p>The reach time in a state is the inverse of the values of the stationary law.<\/p><p>When we calculate Q\u00b2 and Q<sup>3<\/sup>, we notice that the value in index (1,1) is positive, so there is no period. It is not useful to check the whole diagonal because we have shown that all the states are part of the same class.<\/p><p>The chain is ergodic, its asymptotic character tends towards the stationary distribution.<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-365ced4 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"365ced4\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e48cf71\" data-id=\"e48cf71\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-476c440 elementor-widget-divider--view-line elementor-widget elementor-widget-divider\" data-id=\"476c440\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"divider.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-divider\">\n\t\t\t<span class=\"elementor-divider-separator\">\n\t\t\t\t\t\t<\/span>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-3bf9672 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3bf9672\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-80e592a\" data-id=\"80e592a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b366b3e elementor-widget elementor-widget-heading\" data-id=\"b366b3e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Exercice-9\"><\/span>Exercise 9<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2d941d0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2d941d0\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-b41e6fd\" data-id=\"b41e6fd\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5e80450 elementor-widget elementor-widget-text-editor\" data-id=\"5e80450\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>To represent the passage of a phosphorus molecule in an ecosystem, we will consider four possible states:<\/p><ol><li>the molecule is in the soil,<\/li><li>the molecule is in the grass,<\/li><li>the molecule was taken up by cattle,<\/li><li>the molecule has left the ecosystem.<\/li><\/ol><p>The transition matrix is as follows:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10522 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image93.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"140\" height=\"130\" title=\"\"><\/p><p>Represent the associated Markov chain. Study the Markov chain.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-203b787 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"203b787\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-929086b\" data-id=\"929086b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2f496ec elementor-widget elementor-widget-toggle\" data-id=\"2f496ec\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-4951\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-4951\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Solution<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-4951\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-4951\"><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10523 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image94.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"260\" height=\"207\" title=\"\"><\/p><p>Pi = (0, 0, 0, 1) because of the absorbent state. It is possible to calculate N and B by:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10524 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image95.png\" alt=\"exercises corrected on discrete-time Markov chains and asymptotic behavior, class, ergodic chain, absorption.\" width=\"383\" height=\"240\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image95.png 383w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/10\/Image95-300x188.png 300w\" sizes=\"(max-width: 383px) 100vw, 383px\" \/><\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>","protected":false},"excerpt":{"rendered":"<p>Stochastic process Home page Wiki Corrected exercises: Markov chains in discrete time This page gathers many corrected exercises on chains \u2026 <\/p>","protected":false},"author":1,"featured_media":0,"parent":5007,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-10482","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/10482","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/comments?post=10482"}],"version-history":[{"count":15,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/10482\/revisions"}],"predecessor-version":[{"id":20552,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/10482\/revisions\/20552"}],"up":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/5007"}],"wp:attachment":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/media?parent=10482"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}