{"id":435,"date":"2015-01-05T20:29:37","date_gmt":"2015-01-05T20:29:37","guid":{"rendered":"http:\/\/frees.pajarel.net\/?page_id=435"},"modified":"2015-02-20T19:15:19","modified_gmt":"2015-02-21T01:15:19","slug":"discrete-distribution-need-translation-for-eqnarray","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/interest-rate-risks-and-simulation\/4-simulation\/4-2-inverse-transform-image-inversedf-needed\/discrete-distribution-need-translation-for-eqnarray\/","title":{"rendered":"Discrete Distribution"},"content":{"rendered":"<p><strong>Discrete Distribution Example.<\/strong> Consider the time of a machine failure in the first five years. The distribution of failure times is given as:<\/p>\n<table>\n<tr>\n<td>Time (\\(x\\))<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>Probability<\/td>\n<td>0.1<\/td>\n<td>0.2<\/td>\n<td>0.1<\/td>\n<td>0.4<\/td>\n<td>0.2<\/td>\n<\/tr>\n<tr>\n<td>\\(F(x)\\)<\/td>\n<td>0.1<\/td>\n<td>0.3<\/td>\n<td>0.4<\/td>\n<td>0.8<\/td>\n<td><\/td>\n<\/tr>\n<\/table>\n<p><strong>Figure 2. Discrete Distribution Function<\/strong><\/p>\n<p><a href=\"http:\/\/www.ssc.wisc.edu\/~jfrees\/wp-content\/uploads\/2015\/01\/DiscreteRVDF.eps_.jpg\"><img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.ssc.wisc.edu\/~jfrees\/wp-content\/uploads\/2015\/01\/DiscreteRVDF.eps_-300x188.jpg\" alt=\"DiscreteRVDF\" width=\"300\" height=\"188\" class=\"aligncenter size-medium wp-image-436\" srcset=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-content\/uploads\/2015\/01\/DiscreteRVDF.eps_-300x188.jpg 300w, https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-content\/uploads\/2015\/01\/DiscreteRVDF.eps_-1024x640.jpg 1024w, https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-content\/uploads\/2015\/01\/DiscreteRVDF.eps_.jpg 1152w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p>Using the graph of the distribution function, with the inverse transform we may define<br \/>\n\\begin{eqnarray*}<br \/>\nX = \\left\\{ \\begin{array}{cc}<br \/>\n1 &#038; 0 \\lt U \\lt 0.1 \\\\<br \/>\n2 &#038; 0.1 \\leq U \\lt 0.3\\\\<br \/>\n3 &#038; 0.3 \\leq U \\lt 0.4\\\\<br \/>\n4 &#038; 0.4 \\leq U \\lt 0.8 \\\\<br \/>\n5 &#038; 0.8 \\leq U \\lt 1.0 .<br \/>\n\\end{array} \\right.<br \/>\n\\end{eqnarray*}<br \/>\nFor general discrete random variables, there may not be an ordering of outcomes. For example, a person could own one of five types of life insurance products and we might use the following algorithm to generate random outcomes:<br \/>\n\\begin{eqnarray*}<br \/>\nX = \\left\\{ \\begin{array}{cc}<br \/>\n\\textrm{whole life} &#038; 0 < U < 0.1 \\\\\n\\textrm{endowment} &#038; 0.1 \\leq U < 0.3\\\\\n\\textrm{term life} &#038; 0.3 \\leq U < 0.4\\\\\n\\textrm{universal life} &#038; 0.4 \\leq U < 0.8 \\\\\n\\textrm{variable life} &#038; 0.8 \\leq U < 1.0 .\n\\end{array} \\right.\n\\end{eqnarray*}\nAnother analyst may use an alternative procedure such as:\n\\begin{eqnarray*}\nX = \\left\\{\n\\begin{array}{cc}\n\\textrm{whole life} &#038; 0.9 < U < 1.0 \\\\\n\\textrm{endowment} &#038; 0.7 \\leq U < 0.9\\\\\n\\textrm{term life} &#038; 0.6 \\leq U < 0.7\\\\\n\\textrm{universal life} &#038; 0.2 \\leq U < 0.6 \\\\\n\\textrm{variable life} &#038; 0 \\leq U < 0.2 .\n\\end{array} \\right.\n\\end{eqnarray*}\nBoth algorithms produce (in the long-run) the same probabilities, e.g., \\(Pr(\\textrm{whole life})=0.1\\), and so forth. So, neither is incorrect. You should be aware that there is \"more than one way to skin a cat.\" (What an old expression!) Similarly, you could use an alternative algorithm for ordered outcomes (such as failure times 1, 2, 3, 4, or 5, above).\n\n[previous][next]\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Discrete Distribution Example. Consider the time of a machine failure in the first five years. The distribution of failure times is given as: Time (\\(x\\)) 1 2 3 4 5 Probability 0.1 0.2 0.1 0.4 &hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":420,"menu_order":1,"comment_status":"closed","ping_status":"open","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-71","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/435"}],"collection":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/comments?post=435"}],"version-history":[{"count":6,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/435\/revisions"}],"predecessor-version":[{"id":837,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/435\/revisions\/837"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/420"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=435"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}