{"id":380,"date":"2016-10-28T07:11:49","date_gmt":"2016-10-28T12:11:49","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/estg1003\/?p=380"},"modified":"2026-04-05T16:42:30","modified_gmt":"2026-04-05T21:42:30","slug":"s1eva2017ti_t1-cadena-de-markov-desde-matriz","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/algoritmos101\/stp-ejemplos\/s1eva2017ti_t1-cadena-de-markov-desde-matriz\/","title":{"rendered":"s1Eva2017TI_T1 Cadena de Markov desde matriz"},"content":{"rendered":"\n<p><strong>Ejercicio<\/strong>: <a href=\"https:\/\/blog.espol.edu.ec\/algoritmos101\/stp-1eva\/1eva2017ti_t1-cadena-de-markov-desde-matriz\/\" data-type=\"post\" data-id=\"22006\">1Eva2017TI_T1 Cadena de Markov desde matriz<\/a><\/p>\n\n\n\n<p><strong>Tema 1 <\/strong><\/p>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> \\begin{pmatrix} 1\/2 &amp; 1\/2 &amp; 0 &amp; 0 \\\\ 9\/10 &amp; 0 &amp; 1\/10 &amp; 0 \\\\ 0 &amp; 1\/10&amp; 0 &amp; 9\/10\\\\ 0 &amp; 0 &amp; 1\/2 &amp; 1\/2 \\end{pmatrix} <\/span>\n\n\n\n<p>Diagrama de Estados de transici\u00f3n:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"555\" height=\"122\" src=\"http:\/\/blog.espol.edu.ec\/algoritmos101\/files\/2017\/06\/1raEva_IT2017_Tema1Diagrama.png\" alt=\"1raEva_IT2017 Tema1 Diagrama\" class=\"wp-image-440\" \/><\/figure>\n\n\n\n<p>Resoluci\u00f3n planteando las ecuaciones<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\u03c0<sub>0<\/sub> = (1\/2)\u03c0<sub>0<\/sub> + (9\/10)\u03c0<sub>1<\/sub>\n\u03c0<sub>1<\/sub> = (1\/2)\u03c0<sub>0<\/sub> + (1\/10)\u03c0<sub>2<\/sub>\n\u03c0<sub>2<\/sub> = (1\/10)\u03c0<sub>1<\/sub> + (1\/2)\u03c0<sub>3<\/sub>\n\u03c0<sub>3<\/sub> = (9\/10)\u03c0<sub>2<\/sub> + (1\/2)\u03c0<sub>3<\/sub>\n\u03c0<sub>0<\/sub> + \u03c0<sub>1<\/sub> + \u03c0<sub>2<\/sub> + \u03c0<sub>3<\/sub> = 1\n\nusando ecuacion (1)\n\u03c0<sub>0<\/sub> -(1\/2)\u03c0<sub>0<\/sub> = (9\/10)\u03c0<sub>1<\/sub>\n(1\/2)\u03c0<sub>0<\/sub>  = (9\/10)\u03c0<sub>1<\/sub>\n\u03c0<sub>1<\/sub> = (5\/9)\u03c0<sub>0<\/sub>\n\nusando ecuacion(2)\n(5\/9)\u03c0<sub>0<\/sub> = (1\/2)\u03c0<sub>0<\/sub> + (1\/10)\u03c0<sub>2<\/sub>\n(5\/9)\u03c0<sub>0<\/sub> - (1\/2)\u03c0<sub>0<\/sub> = (1\/10)\u03c0<sub>2<\/sub>\n&#091;(10 - 9)\/18] \u03c0<sub>0<\/sub> = (1\/10)\u03c0<sub>2<\/sub>\n&#091;1\/18] \u03c0<sub>0<\/sub> = (1\/10)\u03c0<sub>2<\/sub>\n\u03c0<sub>2<\/sub> = (10\/18) \u03c0<sub>0<\/sub>\n\u03c0<sub>2<\/sub> = (5\/9) \u03c0<sub>0<\/sub>\n\nusando ecuacion (3)\n(5\/9) \u03c0<sub>0<\/sub> = (1\/10)(5\/9)\u03c0<sub>0<\/sub> + (1\/2)\u03c0<sub>3<\/sub>\n(5\/9) \u03c0<sub>0<\/sub> - (1\/10)(5\/9)\u03c0<sub>0<\/sub> = (1\/2)\u03c0<sub>3<\/sub>\n(1-1\/10)(5\/9) \u03c0<sub>0<\/sub> = (1\/2)\u03c0<sub>3<\/sub>\n(9\/10)(5\/9) \u03c0<sub>0<\/sub> = (1\/2)\u03c0<sub>3<\/sub>\n(1\/2) \u03c0<sub>0<\/sub> = (1\/2)\u03c0<sub>3<\/sub>\n\u03c0<sub>3<\/sub> = \u03c0<sub>0<\/sub>\n\nusando la ecuaci\u00f3n (5)\n\u03c0<sub>0<\/sub> + (5\/9)\u03c0<sub>0<\/sub> + (5\/9) \u03c0<sub>0<\/sub> +  \u03c0<sub>0<\/sub> = 1\n(1 + 5\/9 + 5\/9 + 1) \u03c0<sub>0<\/sub> = 1\n(2 + 10\/9) \u03c0<sub>0<\/sub> = 1\n(28\/9) \u03c0<sub>0<\/sub> = 1\n\u03c0<sub>0<\/sub> = (9\/28)\n\n\u03c0<sub>1<\/sub> = (5\/9)(9\/28) = 5\/28\n\u03c0<sub>2<\/sub> = 5\/28\n\u03c0<sub>3<\/sub> = 9\/28<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<p>resoluci\u00f3n usando Numpy de Python<\/p>\n\n\n<div class=\"wp-block-syntaxhighlighter-code alignwide\"><pre class=\"brush: python; title: ; notranslate\" title=\"\">\n# Tema 1. matriz de transici\u00f3n\nimport numpy as np\n\np=np.array(&#x5B;\n    &#x5B; 1\/2, 1\/2,   0,   0],\n    &#x5B;9\/10,   0,1\/10,   0],\n    &#x5B;   0,1\/10,   0,9\/10],\n    &#x5B;   0,   0, 1\/2, 1\/2]\n    ])\nn=200\npn=np.linalg.matrix_power(p,n)\nprint(pn)\n\n# Resolviendo por matrices A= A&lt;sup&gt;T&lt;\/sup&gt;-I) y el vector de ceros terminado en 1\nk=len(p)\nA=p.transpose()\nA=A-np.identity(k, dtype=int)\n# la \u00faltima fila se sustituye por la suma de probabilidades\nA&#x5B;-1,:]=np.ones(k,dtype=int)\nB=np.zeros(k,dtype=int)\nB&#x5B;-1]=1  # el \u00faltimo\nPncalc=np.linalg.solve(A,B)\nprint('largo plazo')\nprint(Pncalc)\n<\/pre><\/div>\n\n\n<pre class=\"wp-block-code alignwide\"><code>&#091;&#091; 0.32142907  0.17857167  0.17857119  0.32142808]\n &#091; 0.321429    0.17857164  0.17857122  0.32142814]\n &#091; 0.32142814  0.17857122  0.17857164  0.321429  ]\n &#091; 0.32142808  0.17857119  0.17857167  0.32142907]]\nlargo plazo\n&#091; 0.32142857  0.17857143  0.17857143  0.32142857]<\/code><\/pre>\n\n\n\n<p>Verificando que las fracciones sean los valores encontrados por Python:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>print(5\/28)<\/code><\/pre>\n\n\n\n<pre class=\"wp-block-code\"><code>0.17857142857142858<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Ejercicio: 1Eva2017TI_T1 Cadena de Markov desde matriz Tema 1 Diagrama de Estados de transici\u00f3n: Resoluci\u00f3n planteando las ecuaciones resoluci\u00f3n usando Numpy de Python Verificando que las fracciones sean los valores encontrados por Python:<\/p>\n","protected":false},"author":8043,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"wp-custom-template-entrada-stp-ejercicios","format":"standard","meta":{"footnotes":""},"categories":[203],"tags":[58,237],"class_list":["post-380","post","type-post","status-publish","format-standard","hentry","category-stp-ejemplos","tag-ejemplos-python","tag-pestocasticos"],"_links":{"self":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/380","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/users\/8043"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/comments?post=380"}],"version-history":[{"count":4,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/380\/revisions"}],"predecessor-version":[{"id":23545,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/380\/revisions\/23545"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/media?parent=380"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/categories?post=380"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/tags?post=380"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}