STAT3801_2011Unit2[1]

STAT3801_2011Unit2[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 2 2010 - 2011 2 nd Semester Multiple life functions There must be a finite set of members for each member define future lifetime random variable Status survival can be determined at any future time 2.1 Joint distribution of future lifetimes Example 9.2.1 f T ( x ) T ( y ) ( s, t ) = ( 0 . 0006( t - s ) 2 0 < s < 10 , 0 < t < 10 0 elsewhere Required to determine Joint d. f. of T ( x ) and T ( y ) Marginal p. d. f., d. f., s p x and μ ( x + s ) for T ( x ) Correlation coefficient of T ( x ) and T ( y )
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2 Sample space of T ( x ) and T ( y ) 6 - 0 T ( x ) T ( y ) I II III IV 10 10
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3 Region I F T ( x ) T ( y ) ( s, t ) = Pr [ T ( x ) s and T ( y ) t ] = Z s -∞ Z t -∞ f T ( x ) T ( y ) ( u, v ) dv du = Z s 0 Z t 0 0 . 0006( v - u ) 2 dv du = Z s 0 0 . 0002[( t - u ) 3 + u 3 ] du = 0 . 00005[ s 4 + t 4 - ( t - s ) 4 ] 0 < s 10 , 0 < t 10 Regions II, III and IV F T ( x ) T ( y ) ( s, t ) = F T ( x ) T ( y ) ( s, 10) = F T ( x ) ( s ) = 1 2 + 0 . 00005[ s 4 - (10 - s ) 4 ] in Region II = F T ( x ) T ( y ) (10 , t ) = F T ( y ) ( t ) = 1 2 + 0 . 00005[ t 4 - (10 - t ) 4 ] in Region IV = 1 in Region III Marginal distribution and density of T ( x ) F T ( x ) T ( y ) ( s, 10) = F T ( x ) ( s ) = 0 s 0 1 2 + 0 . 00005[ s 4 - (10 - s ) 4 ] 0 < s 10 1 s > 10 f T ( x ) ( s ) = F 0 T ( x ) ( s ) = ( 0 . 0002[ s 3 + (10 - s ) 3 ] 0 < s < 10 0 elsewhere
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4 Survival probability and force of mortality s p x = 1 - F T ( x ) ( s ) = 1 2 + 0 . 00005[(10 - s ) 4 - s 4 ] 0 < s 10 = 0 s > 10 μ ( x + t ) = f T ( x ) ( t ) 1 - F T ( x ) ( t ) = 0 . 0002[ s 3 + (10 - s ) 3 ] 1 2 + 0 . 00005[(10 - s ) 4 - s 4 ] 0 < s 10 Correlation coefficient of T ( x ) and T ( y ) E [ T ( x )] = Z 10 0 s (0 . 0002)[ s 3 + (10 - s ) 3 ] ds = 5 = E [ T ( y )] E [ T ( x ) 2 ] = Z 10 0 s 2 (0 . 0002)[ s 3 + (10 - s ) 3 ] ds = 110 3 = E [ T ( y ) 2 ] V ar [ T ( x )] = 35 3 = V ar [ T ( y )] E [ T ( x ) T ( y )] = Z 10 0 Z 10 0 s t (0 . 0006)( t - s ) 2 ds dt = 50 3 Cov [ T ( x ) , T ( y )] = E [ T ( x ) T ( y )] - E [ T ( x )] E [ T ( y )] = - 25 3 ρ T ( x ) T ( y ) = Cov [ T ( x ) , T ( y )] σ T ( x ) σ T ( y ) = - 25 3 35 3 = - 5 7
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5 2.2 Joint survival function s T ( x ) T ( y ) ( s, t ) = Pr [ T ( x ) > s and T ( y ) > t ] Sample space of future lifetime random variables T ( x ) and T ( y ) 6 - 0 T ( x ) T ( y ) Region A ( s, t ) @ @ @ R Region B s t Joint survival function For 0 < s < 10 and 0 < t < 10 , s T ( x ) T ( y ) ( s, t ) = Pr [ T ( x ) > s T ( y ) > t ] = Z s Z t f T ( x ) T ( y ) ( u, v ) dv du = Z 10 s Z 10 t 0 . 0006( v - u ) 2 dv du = Z 10 s 0 . 0002[(10 - u ) 3 - ( t - u ) 3 ] du = 0 . 00005[(10
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STAT3801_2011Unit2[1] - THE UNIVERSITY OF HONG KONG...

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