Notes-Ch9-331W09 - Review Notes for Chapter 9 ACTSC 331...

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Review Notes for Chapter 9 – ACTSC 331, Winter Spring 2009 1. The joint distributions of future lifetimes: Let T ( x ) and T ( y ) denote the future lifetimes of ( x ) and ( y ), respectively. Then T ( x ) and T ( y ) are non-negative continuous random variables. (a) The joint distribution function of T ( x ) and T ( y ): F T ( x ) T ( y ) ( s, t ) = Pr { T ( x ) s, T ( y ) t } . (b) The joint survival function of T ( x ) and T ( y ): s T ( x ) T ( y ) ( s, t ) = Pr { T ( x ) > s, T ( y ) > t } . (c) The joint density function of T ( x ) and T ( y ): f T ( x ) T ( y ) ( s, t ) = 2 ∂s ∂t F T ( x ) T ( y ) ( s, t ) = 2 ∂s ∂t s T ( x ) T ( y ) ( s, t ) . (d) The relationships between the joint density function and the joint distribution and survival functions: F T ( x ) T ( y ) ( s, t ) = s -∞ t -∞ f T ( x ) T ( y ) ( u, v ) dvdu, s T ( x ) T ( y ) ( s, t ) = s t f T ( x ) T ( y ) ( u, v ) dvdu. (e) The covariance of T ( x ) and T ( y ): Cov[ T ( x ) , T ( y )] = E[ T ( x ) T ( y )] - E [ T ( x )] E [ T ( y )] . (f) The correlation coefficient of T ( x ) and T ( y ): ρ T ( x ) T ( y ) = Cov [ T ( x ) ,T ( y )] Var [ T ( x )] Var [ T ( y )] . (g) If T ( x ) and T ( y ) are independent, then f T ( x ) T ( y ) ( s, t ) = f T ( x ) ( s ) f T ( y ) ( t ) , F T ( x ) T ( y ) ( s, t ) = F T ( x ) ( s ) F T ( y ) ( t ) , s T ( x ) T ( y ) ( s, t ) = s T ( x ) ( s ) s T ( y ) ( t ) . 2. Review of calculations of E ( X ) and E ( X 2 ) : (a) If X is a nonnegative continuous random variable with density function f X ( t ) and survival function s X ( t ), then E ( X ) = 0 s X ( t ) dt = 0 t f X ( t ) dt, E ( X 2 ) = 2 0 t s X ( t ) dt = 0 t 2 f X ( t ) dt. (b) If X is a nonnegative integer-valued random variable, then E ( X ) = n =0 Pr { X > n } = n =0 n Pr { X = n } . 1
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3. The Joint-Life Status: The joint-life status of ( x ) and ( y ) is denoted by ( xy ). The future lifetime of the joint-life status is denoted by T ( xy ) = min { T ( x ) , T ( y ) } , which is also called the time-until-failure of the join-life status. T ( xy ) is the time of the first death of ( x ) and ( y ). (a) The distribution function of T ( xy ): t q xy = Pr { T ( xy ) t } = Pr { ( T ( x ) t ) ( T ( y ) t ) } = t q x + t q y - F T ( x ) T ( y ) ( t, t ) . (b) The survival function of T ( xy ): t p xy = 1 - t q xy = Pr { T ( xy ) > t } = Pr { T ( x ) > t, T ( y ) > t } = s T ( x ) T ( y ) ( t, t ) . (c) If T ( x ) and T ( y ) are independent, then t p xy = t p x t p y and t q xy = 1 - t p x t p y = t q x + t q y - t q x t q y . (d) The complete expectation of the joint-life status or the expected time of the first death: e xy = E[ T ( xy )] = 0 t p xy dt. (e) The complete variance of the joint-life status or the variance of the time of the first death: Var[ T ( xy )] = E[ T 2 ( xy )] - (E[ T ( xy )]) 2 = 2 0 t t p xy dt - ( e xy ) 2 .
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