Start with a generic time until failure \\(T\\) that may be a function of one or more lives.<\/p>\n
Define the curtate time until failure \\(K = [T]\\), where \\([\\cdot]\\) denotes the greatest integer function. Then,
\n\\begin{eqnarray*}
\nPr(K=k) = Pr( k \\le T < k+1) = F_T(k+1) - F_T(k).\n\\end{eqnarray*}\nSpecial Case. Joint Life Status. In this case \\(T=T(xy)\\), and so \\(K\\) may be denoted as \\(K(xy) = [T(xy)]\\). We may also think of the curtate random variable as \\(K(xy) = \\min(K(x), K(y))\\).\n\nThis has probability mass function\n\\begin{eqnarray*}\nPr(K=k) &=& ~_{k+1} q_{xy} - ~ _k q_{xy} = ~ _k p_{xy} - ~_{k+1} p_{xy}\\\n& =& ~ _k p_{xy} q_{x+k:y+k} \\equiv ~_{k|} q_{xy} .\n\\end{eqnarray*}\nFor insurances, we have the first-to-die insurance\n\\begin{eqnarray*}\nA_{xy} = \\mathrm{E~} v^{K(xy)+1} = \\sum_{k=0}^{\\infty} v^{k+1}\n~_{k|} q_{xy} .\n\\end{eqnarray*}\nThis is the EPV for a payment of 1 at the end of the year of the first death among \\(x\\) and \\(y\\).\nFor annuities, we have\n\\begin{eqnarray*}\n\\ddot{a}_{xy} = \\mathrm{E~} \\ddot{a}_{\\overline{K(xy)+1}|} =\n\\sum_{k=0}^{\\infty} v^k ~ _k p_{xy} .\n\\end{eqnarray*}\nThis is the EPV for a payment of 1 at the beginning of each year while both \\(x\\) and \\(y\\) are alive.\n\n[previous][next]\n<\/p>\n","protected":false},"excerpt":{"rendered":"
Start with a generic time until failure \\(T\\) that may be a function of one or more lives. Define the curtate time until failure \\(K = [T]\\), where \\([\\cdot]\\) denotes the greatest integer function. Then, …<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":90,"menu_order":2,"comment_status":"closed","ping_status":"open","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-1C","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/100"}],"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=100"}],"version-history":[{"count":5,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/100\/revisions"}],"predecessor-version":[{"id":1529,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/100\/revisions\/1529"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/90"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=100"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}