STAT3801_2011Unit2b[1]

STAT3801_2011Unit2b[1] - THE UNIVERSITY OF HONG KONG...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT3801 ADVANCED LIFE CONTINGENCIES Unit 2b 2010-11 2 nd semester 2.12 Simple contingent functions In addition to being independent on time of failure of status Contingent on order of deaths of individuals in the group Assume T ( x ) and T ( y ) are continuous Probability that ( x ) dies before ( y ) and before n years from now n q 1 xy = Pr [ T ( x ) < T ( y ) , T ( x ) < n ] = Z n 0 Z s f T ( x ) T ( y ) ( s, t ) dt ds = Z n 0 Z s f T ( y ) | T ( x ) ( t | s ) dt f T ( x ) ( s ) ds = Z n 0 Pr [ T ( y ) > s | T ( x ) = s ] f T ( x ) ( s ) ds = Z n 0 Pr [ T ( y ) > s | T ( x ) = s ] s p x μ ( x + s ) ds
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2 For independent case Pr [ T ( y ) > s | T ( x ) = s ] = Pr [ T ( y ) > s ] = s p y n q 1 xy = Z n 0 s p y s p x μ ( x + s ) ds = Z n 0 s p xy μ ( x + s ) ds Example 2.19: Calculate 5 q 1 xy when f T ( x ) T ( y ) ( s, t ) = ( 0 . 0006( t - s ) 2 0 < s < 10 , 0 < t < 10 0 elsewhere Solution 5 q 1 xy = Z 5 0 Z 10 s 0 . 0006( t - s ) 2 dt ds = Z 5 0 £ 0 . 0002( t - s ) 3 / t =10 t = s ds = Z 5 0 0 . 0002(10 - s ) 3 ds = 0 . 46875
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3 Probability that ( y ) dies after ( x ) and before n years from now n q 2
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STAT3801_2011Unit2b[1] - THE UNIVERSITY OF HONG KONG...

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