{"id":431,"date":"2015-01-05T20:18:00","date_gmt":"2015-01-05T20:18:00","guid":{"rendered":"http:\/\/frees.pajarel.net\/?page_id=431"},"modified":"2015-02-20T19:14:34","modified_gmt":"2015-02-21T01:14:34","slug":"bernoulli-distribution","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\/bernoulli-distribution\/","title":{"rendered":"Bernoulli Distribution"},"content":{"rendered":"<p><strong>Bernoulli Distribution Example.<\/strong> Suppose that we wish to simulate random variables from a Bernoulli distribution with parameter \\(p=0.85\\). Consider a graph of the cumulative distribution function.<\/p>\n<p><strong>Figure 1. Bernoulli Distribution Function<\/strong><\/p>\n<p><a href=\"http:\/\/www.ssc.wisc.edu\/~jfrees\/wp-content\/uploads\/2015\/01\/BinaryDF.eps_.jpg\"><img decoding=\"async\" loading=\"lazy\" src=\"http:\/\/www.ssc.wisc.edu\/~jfrees\/wp-content\/uploads\/2015\/01\/BinaryDF.eps_-300x188.jpg\" alt=\"BinaryDF\" width=\"300\" height=\"188\" class=\"aligncenter size-medium wp-image-432\" srcset=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-content\/uploads\/2015\/01\/BinaryDF.eps_-300x188.jpg 300w, https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-content\/uploads\/2015\/01\/BinaryDF.eps_-1024x640.jpg 1024w, https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-content\/uploads\/2015\/01\/BinaryDF.eps_.jpg 1152w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p>This shows that the quantile function can be written as<br \/>\n\\begin{eqnarray*}<br \/>\nF^{-1}(y) = \\left\\{<br \/>\n\\begin{array}{cr}<br \/>\n0 &#038; 0 < y < 0.85 \\\\\n1 &#038; 0.85 \\leq y < 1.0 \\\\\n\\end{array} \\right.\n\\end{eqnarray*}\nThus, with the inverse transform we may define\n\\begin{eqnarray*}\nX = \\left\\{ \\begin{array}{cc}\n0 &#038; 0 < U < 0.85 \\\\\n1 &#038; 0.85 \\leq U < 1.0\n\\end{array} \\right.\n\\end{eqnarray*}\n<em>Some Numbers.<\/em> Generate three random numbers to get<\/p>\n<table>\n<tr>\n<td>\\(U\\)<\/td>\n<td>0.26321364<\/td>\n<td>0.196884752<\/td>\n<td>0.897884218<\/td>\n<\/tr>\n<tr>\n<td>\\(X=F^{-1}(U)\\)<\/td>\n<td>0<\/td>\n<td>0<\/td>\n<td>9.909071325<\/td>\n<\/tr>\n<\/table>\n<p><div class=\"alignleft\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/interest-rate-risks-and-simulation\/4-simulation\/4-2-inverse-transform-image-inversedf-needed\/\" title=\"4.2 Inverse Transform\">&#9668 Previous page<\/a><\/div><div class=\"alignright\"><a href=\"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=\"Discrete Distribution\">Next page &#9658<\/a><\/div><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Bernoulli Distribution Example. Suppose that we wish to simulate random variables from a Bernoulli distribution with parameter \\(p=0.85\\). Consider a graph of the cumulative distribution function. Figure 1. Bernoulli Distribution Function This shows that the &hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":420,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-6X","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/431"}],"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=431"}],"version-history":[{"count":6,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/431\/revisions"}],"predecessor-version":[{"id":1650,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/431\/revisions\/1650"}],"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=431"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}