{"id":71,"date":"2014-12-26T13:00:44","date_gmt":"2014-12-26T13:00:44","guid":{"rendered":"http:\/\/frees.pajarel.net\/?page_id=71"},"modified":"2015-02-20T15:47:50","modified_gmt":"2015-02-20T21:47:50","slug":"exercises","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/classic-joint-life-models\/1-joint-life-fundamentals\/exercises\/","title":{"rendered":"Joint Probability Exercises"},"content":{"rendered":"

Exercise 1<\/strong>. Suppose that \\(T(x)\\) is uniformly distributed (DeMoivre) over \\((0,w_x-x)\\), \\(T(y)\\) is uniformly distributed over \\((0,w_y-y)\\), and that \\(T(x)\\) and \\(T(y)\\) are independent. Let \\(w_x=100\\), \\(x=30\\), \\(w_y=110\\), and \\(y=28\\). Determine:<\/p>\n

\n
– Soln<\/div>\n
+ Soln<\/div>\n
Solution: For \\(T(x) \\sim U(0,w_x-x)\\) (DeMoivre), recall that
\n\\begin{eqnarray*}
\n~_t p_x = 1 -\\frac{t}{w_x-x}, ~_t p_x \\mu_{x+t} = \\frac{1}{w_x-x}, ~\\textrm{and}~ \\mu_{x+t} = \\frac{1}{w_x-x-t}.
\n\\end{eqnarray*}
\nThus,
\n\\begin{eqnarray*}
\n~_{20} p_{30:28} &=& ~_{20} p_{30}^m ~_{20} p_{28}^f = \\left(1-\\frac{20}{100-30}\\right)
\n\\left(1-\\frac{20}{110-28}\\right) = 0.587.
\n\\end{eqnarray*}<\/div>\n<\/div>\n

a) \\(~_{20} p_{30:28}\\)<\/p>\n

<\/p>\n

\n
– Soln<\/div>\n
+ Soln<\/div>\n
 
\n\\begin{eqnarray*}
\n\\mu_{30:28}(20) = \\mu_{30+20}^m + \\mu_{28+20}^f = \\frac{1}{100-50}+\\frac{1}{110-48} = 0.0306. \\end{eqnarray*}<\/div>\n<\/div>\n

b) \\( \\mu_{30:28}(20)\\)<\/p>\n

<\/p>\n

\n
– Soln<\/div>\n
+ Soln<\/div>\n
 
\n\\begin{eqnarray*}
\n~_{20} p_{\\overline{30:28}} & =&
\n~_{20} p_{30}^m + ~_{20} p_{28}^f – ~_{20} p_{30:28} \\
\n& =& \\frac{50}{70} + \\frac{62}{82} – 0.587 = 0.9457
\n\\end{eqnarray*}<\/div>\n<\/div>\n

c) \\(~_{20} p_{\\overline{30:28}}\\)<\/p>\n

<\/p>\n

\n

Exercise 2<\/strong>. Suppose that \\(T(x)\\) is exponentially distributed with force of mortality \\(\\mu_x(t)=\\mu_1=0.02\\), \\(T(y)\\) is exponentially distributed with force of mortality \\(\\mu_y(t)=\\mu_2=0.015\\), and that \\(T(x)\\) and \\(T(y)\\) are independent.<\/p>\n

Provide expressions for:<\/p>\n

\n
– Soln<\/div>\n
+ Soln<\/div>\n
For \\(T(x)\\) is exponentially distributed with force of mortality \\(\\mu_x(t)=\\mu_1\\), recall that
\n\\begin{eqnarray*}
\n~_t p_x = \\exp(-\\mu_1 t).
\n\\end{eqnarray*}
\nThus,
\n\\begin{eqnarray*}
\n~_t p_{xy} &=& ~_t p_x \\times ~_t p_y = \\exp(-\\mu_1 t)\\exp(-\\mu_2 t) \\\\
\n&=& \\exp(-(0.02+0.15)t) = \\exp(-0.035 t).
\n\\end{eqnarray*}\n<\/div>\n<\/div>\n

a) \\(~_t p_{xy}\\)<\/p>\n

<\/p>\n

\n
– Soln<\/div>\n
+ Soln<\/div>\n
 
\n\\begin{eqnarray*}
\n\\mu_{xy}(t) = \\mu_{x}(t)^m+\\mu_{y}(t)^f = \\mu_1 +\\mu_2 = 0.035.
\n\\end{eqnarray*}\n<\/div>\n<\/div>\n

b) \\( \\mu_{xy}(t)\\)<\/p>\n

<\/p>\n

\n
– Soln<\/div>\n
+ Soln<\/div>\n
 
\n\\begin{eqnarray*}
\n~_t p_{\\overline{xy}} &=& ~_t p_x + ~_t p_y – ~_t p_x ~ ~_t p_y \\\\
\n&=& \\exp(-\\mu_1 t) + \\exp(-\\mu_2 t) – \\exp(-(\\mu_1+\\mu_2) t) \\\\
\n&=& \\exp(-0.02 t) + \\exp(-0.015 t) – \\exp(-0.035 t) .
\n\\end{eqnarray*}\n<\/div>\n<\/div>\n

c) \\(~_t p_{\\overline{xy}}\\)<\/p>\n

<\/p>\n

\n

Exercise 3<\/strong>. Recall that we can express the Gompertz force of mortality as \\(\\mu_x = B c^x\\). In this case, we may express the survival function as \\( S(x) = \\exp(-m(c^x-1))\\), where \\(m=B\/ \\ln c\\) and the conditional survivor function for \\(T(x)\\) as
\n\\begin{eqnarray*}
\n~_t p_x = \\exp(-m(c^{x+t}-c^x))= \\exp(-m c^x(c^t-1)) .
\n\\end{eqnarray*} Assume that the Gompertz force governs mortality for males and females (with a common parameters \\(m\\) and \\(c\\)).<\/p>\n

Determine the value of \\(w\\) so that we may write \\(~_t p_{xy} = ~_t p_w\\). That is, show that we can compute joint life probabilities using a single life table for the Gompertz case.<\/p>\n

\n
– Soln<\/div>\n
+ Soln<\/div>\n
Solution.<\/i> The joint life survival probability is
\n\\begin{eqnarray*}
\n~_t p_{xy} &=& ~_t p_x \\times ~_t p_y \\\\
\n&=& \\exp(-m c^x(c^t-1)) \\exp(-m c^y(c^t-1))\\\\
\n&=& \\exp(-( c^x + c^y) \\times m (c^t-1)) \\\\
\n\\end{eqnarray*}
\nSo, define \\(w\\) to be the solution of \\(c^w = c^x + c^y\\). With this, we may write
\n\\begin{eqnarray*}
\n~_t p_{xy} &=& \\exp(-( c^x + c^y) \\times m (c^t-1)) =\\exp(- m c^w (c^t-1))= ~_t p_w,
\n\\end{eqnarray*}
\nas required.\n<\/div>\n<\/div>\n

\n

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Exercise 1. Suppose that \\(T(x)\\) is uniformly distributed (DeMoivre) over \\((0,w_x-x)\\), \\(T(y)\\) is uniformly distributed over \\((0,w_y-y)\\), and that \\(T(x)\\) and \\(T(y)\\) are independent. Let \\(w_x=100\\), \\(x=30\\), \\(w_y=110\\), and \\(y=28\\). Determine: – Soln + Soln …<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":43,"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-19","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/71"}],"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=71"}],"version-history":[{"count":9,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/71\/revisions"}],"predecessor-version":[{"id":1521,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/71\/revisions\/1521"}],"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=71"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}