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Part-3-331-2010-F

# Part-3-331-2010-F - Summary of Course Notes ACTSC 331 FALL...

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Summary of Course Notes – ACTSC 331, FALL 2010 Part 3 – Multiple Life Functions 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 TxTy ( 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 TxTy ( s, t ) . (d) 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 TxTy ( s, t ) = s Tx ( s ) s Ty ( 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 ) = Z 0 s X ( t ) dt = Z 0 t f X ( t ) dt, E ( X 2 ) = 2 Z 0 t s X ( t ) dt = Z 0 t 2 f X ( t ) dt. (b) If X is a nonnegative integer-valued random variable, then E ( X ) = X n =0 Pr { X > n } = X n =0 n Pr { X = n } . 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 ) . 1

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(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 TxTy ( 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 ] = Z 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 Z 0 t t p xy dt - ( e xy ) 2 . (f) The density function of T xy : f T xy ( t ) = d dt t q xy .
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