{"id":77,"date":"2014-12-26T13:06:11","date_gmt":"2014-12-26T13:06:11","guid":{"rendered":"http:\/\/frees.pajarel.net\/?page_id=77"},"modified":"2015-02-20T16:04:28","modified_gmt":"2015-02-20T22:04:28","slug":"lifetime-moments","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/classic-joint-life-models\/1-joint-life-fundamentals\/lifetime-moments\/","title":{"rendered":"Lifetime Moments"},"content":{"rendered":"

For the expected survivor time of the joint life status, we have
\n\\begin{eqnarray*}
\n\\dot{e}_{xy} = E ~ T(xy) &=& \\int_0^{\\infty} t f_T(xy)(t) dt =
\n\\int_0^{\\infty} (1-
\nF_T(xy)(t)) dt \\\\
\n&=& \\int_0^{\\infty} ~_t p_{xy} dt .
\n\\end{eqnarray*} The third equality comes from an integration by parts.<\/p>\n

Similarly, for the expected last-survivor time,
\n\\begin{eqnarray*}
\n\\dot{e}_{\\overline{xy}} &=& E ~ T(\\overline{xy}) = \\int_0^{\\infty}
\n~_t p_{\\overline{xy}} ~ dt \\\\
\n&=& \\int_0^{\\infty}
\n\\left(~_t p_x + ~_t p_y – ~_t p_{xy}\\right) dt \\\\
\n&=& \\dot{e}_{x}+\\dot{e}_{y}-\\dot{e}_{xy}.
\n\\end{eqnarray*} Note that this relation holds without regard to the independence between lives assumption.
\n

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For the expected survivor time of the joint life status, we have \\begin{eqnarray*} \\dot{e}_{xy} = E ~ T(xy) &=& \\int_0^{\\infty} t f_T(xy)(t) dt = \\int_0^{\\infty} (1- F_T(xy)(t)) dt \\\\ &=& \\int_0^{\\infty} ~_t p_{xy} dt . …<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":43,"menu_order":6,"comment_status":"closed","ping_status":"closed","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-1f","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/77"}],"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=77"}],"version-history":[{"count":3,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/77\/revisions"}],"predecessor-version":[{"id":1016,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/77\/revisions\/1016"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/43"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=77"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}