{"id":481,"date":"2015-01-06T16:34:46","date_gmt":"2015-01-06T16:34:46","guid":{"rendered":"http:\/\/frees.pajarel.net\/?page_id=481"},"modified":"2015-02-20T18:35:35","modified_gmt":"2015-02-21T00:35:35","slug":"example-students-writing-actuarial-exams","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/multiple-decrement-models\/3-multiple-decrement-tables\/example-students-writing-actuarial-exams\/","title":{"rendered":"Example: Students Writing Actuarial Exams"},"content":{"rendered":"<p>Let<\/p>\n<ul>\n<li>\\(p_x^{01}\\) be the probability of attaining Fellowship<\/li>\n<li>\\(p_x^{02}\\) be the probability of exiting the examination system due to other  causes.<\/li>\n<li>Decrements occur at the end of the year\t<\/li>\n<\/ul>\n<table>\n<tr>\n<td>\\(x\\)<\/td>\n<td>\\(p_x^{01}\\)<\/td>\n<td>\\(p_x^{02}\\)<\/td>\n<\/tr>\n<tr>\n<td>21<\/td>\n<td>0.008<\/td>\n<td>0.015<\/td>\n<\/tr>\n<tr>\n<td>22<\/td>\n<td>0.015<\/td>\n<td>0.20<\/td>\n<\/tr>\n<tr>\n<td>23<\/td>\n<td>0.025<\/td>\n<td>\\(_t q_x^{(\\tau)}\\)<\/td>\n<\/tr>\n<\/table>\n<p>After attaining fellowship, the only decrement is mortality with constant force \\(\\mu = 0.04.\\)<br \/>\nCalculate the probability that a student age 21 will be living and a Fellow 3 years later.<br \/>\n<em>Solution.<\/em><\/p>\n<table>\n<tr>\n<td>\\(x\\)<\/td>\n<td>\\(p_x^{01}\\)<\/td>\n<td>\\(p_x^{02}\\)<\/td>\n<td>\\(p_x^{0\\bullet}\\)<\/td>\n<td>\\(p_x^{00}\\)<\/td>\n<td>\\(_n p_x^{00}\\)<\/td>\n<td>\\( n\\)<\/td>\n<\/tr>\n<tr>\n<td>21<\/td>\n<td>0.008<\/td>\n<td>0.015<\/td>\n<td>0.158<\/td>\n<td>0.842<\/td>\n<td>0.842<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>22<\/td>\n<td>0.015<\/td>\n<td>0.20<\/td>\n<td>0.215<\/td>\n<td>0.785<\/td>\n<td>0.661<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>23<\/td>\n<td>0.025<\/td>\n<td>0.25<\/td>\n<td>0.275<\/td>\n<td>0.725<\/td>\n<td>0.479<\/td>\n<td><\/td>\n<\/tr>\n<\/table>\n<p>\\begin{eqnarray*}<br \/>\n&#038;&#038;\\text{Desired probability is}\\\\<br \/>\n&#038;=&#038; p_{21}^{01} p_{22} p_{23} + p_{21}^{00} p_{22}^{01} p_{23} + ~_2p_{21}^{00} p_{23}^{01} \\\\<br \/>\n&#038;=&#038;(0.008) e^{-0.04} e^{-0.04}+ (0.842)(.015) e^{-0.04} + (0.661)(0.025) = 0.036<br \/>\n\\end{eqnarray*}<\/p>\n<p><div class=\"alignleft\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/multiple-decrement-models\/3-multiple-decrement-tables\/\" title=\"3. Multiple Decrement Tables\">&#9668 Previous page<\/a><\/div><div class=\"alignright\"><a href=\"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/multiple-decrement-models\/3-multiple-decrement-tables\/excerpt-from-a-critical-illness-multiple-decrement-table\/\" title=\"Excerpt from a Critical Illness Multiple Decrement Table\">Next page &#9658<\/a><\/div><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let \\(p_x^{01}\\) be the probability of attaining Fellowship \\(p_x^{02}\\) be the probability of exiting the examination system due to other causes. Decrements occur at the end of the year \\(x\\) \\(p_x^{01}\\) \\(p_x^{02}\\) 21 0.008 0.015 &hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":478,"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-7L","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/481"}],"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=481"}],"version-history":[{"count":2,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/481\/revisions"}],"predecessor-version":[{"id":1609,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/481\/revisions\/1609"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/478"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=481"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}