{"id":442,"date":"2015-01-05T20:33:57","date_gmt":"2015-01-06T02:33:57","guid":{"rendered":"http:\/\/frees.pajarel.net\/?page_id=442"},"modified":"2015-02-20T19:16:12","modified_gmt":"2015-02-21T01:16:12","slug":"mixed-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\/mixed-distribution-need-translation-for-eqnarray\/","title":{"rendered":"Mixed Distribution"},"content":{"rendered":"<p><strong>Mixed Distribution Example.<\/strong> Consider a random variable that is 0 with probability 70% and is exponentially distributed with parameter \\(\\theta= 10,000\\) with probability 30%. In practice, this might correspond to a 70% chance of having no insurance claims and a 30% chance of a claim &#8211; if a claim occurs, then it is exponentially distributed. The distribution function is given as<br \/>\n\\begin{eqnarray*}<br \/>\nF(y) = \\left\\{ \\begin{array}{cc}<br \/>\n0 &#038; x <0 \\\\\n1 - 0.3 \\exp(-x\/10000) &#038; x \\ge 0 .\n\\end{array} \\right.\n\\end{eqnarray*}\n<strong>Figure 3. Mixed Distribution Function<\/strong><\/p>\n<p><a href=\"http:\/\/www.ssc.wisc.edu\/~jfrees\/wp-content\/uploads\/2015\/01\/MixedDF.eps_.jpg\"><img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.ssc.wisc.edu\/~jfrees\/wp-content\/uploads\/2015\/01\/MixedDF.eps_-300x188.jpg\" alt=\"MixedDF\" width=\"300\" height=\"188\" class=\"aligncenter size-medium wp-image-443\" srcset=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-content\/uploads\/2015\/01\/MixedDF.eps_-300x188.jpg 300w, https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-content\/uploads\/2015\/01\/MixedDF.eps_-1024x640.jpg 1024w, https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-content\/uploads\/2015\/01\/MixedDF.eps_.jpg 1152w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p>From the graph, we can see that the inverse transform for generating random variables with this distribution function is<br \/>\n\\begin{eqnarray*}<br \/>\nX = F^{-1}(U) = \\left\\{<br \/>\n\\begin{array}{cc}<br \/>\n0 &#038; 0< U< 0.7 \\\\\n-1000 \\ln \\left(\\frac{1-U}{0.3}\\right) &#038; 0.7 \\le U < 1 .\n\\end{array} \\right.\n\\end{eqnarray*}\nAs you have seen, for the discrete and mixed random variables, the key is to draw a graph of the distribution function. This allows you to visualize potential values of the inverse function. \n\n[previous][next]\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Mixed Distribution Example. Consider a random variable that is 0 with probability 70% and is exponentially distributed with parameter \\(\\theta= 10,000\\) with probability 30%. In practice, this might correspond to a 70% chance of having &hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":420,"menu_order":3,"comment_status":"closed","ping_status":"closed","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-78","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/442"}],"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=442"}],"version-history":[{"count":6,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/442\/revisions"}],"predecessor-version":[{"id":1652,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/442\/revisions\/1652"}],"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=442"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}