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Notes-Ch10-11-331W09

Notes-Ch10-11-331W09 - Life Contingencies 1 ACTSC 331...

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Life Contingencies 1 - ACTSC 331, Winter 2009 Review Notes for Chapters 10 and 11 1. The multiple decrement model: Assume that life ( x ) is a member of a group. Let T ( x ) = T denote the time of decrement at which life ( x ) leaves the group. T is a nonnegative continuous random variable. Further, assume that there are m causes of decrement. Let J ( x ) = J be a discrete random variable and let ( J = j ) denote cause j, j = 1 , 2 , ..., m . Then J takes only m possible values of 1 , 2 , ..., m. 2. Distributions of T and J : (a) The marginal p.d.f. of T is denoted by f T ( t ) , t 0 with R 0 f T ( t ) dt = 1 . (b) The marginal p.f. of J is denoted by f J ( j ) = Pr { J = j } , j = 1 , 2 , ..., m with m j =1 f J ( j ) = 1. (c) The joint density function of T and J is denoted by f T,J ( t, j ) , t 0 , j = 1 , 2 , ..., m. (d) Relationships between f T ( t ) , f J ( j ) , and f T,J ( t, j ): f T ( t ) = m X j =1 f T,J ( t, j ) and f J ( j ) = Z 0 f T,J ( t, j ) dt. 3. Probabilities of decrements: (a) The probability of decrement between times a and b due to cause j is Pr { a < T b, J = j } = Z b a f T,J ( t, j ) dt. (b) The probability of decrement between times a and b due to all causes is Pr { a < T b } = m X j =1 Pr { a < T b, J = j } = m X j =1 Z b a f T,J ( t, j ) dt. (c) The probability of decrement before time t due to cause j is denoted by t q ( j ) x and t q ( j ) x = Pr { 0 T t, J = j } = Z t 0 f T,J ( s, j ) ds with d dt t q ( j ) x = f T,J ( t, j ) . Note: t q ( j ) x is not a distribution function. (d) The probability of decrement at any time in the future due to cause j is denoted by q ( j ) x and q ( j ) x = Pr { 0 T < , J = j } = f J ( j ) = Z 0 f T,J ( t, j ) dt. 1
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4. Distributions of T : (a) The distribution function of T is denoted by t q ( τ ) x and t q ( τ ) x = Pr { T t } = F T ( t ) = Z t 0 f T ( s ) ds, which is the probability of decrement before time t due to all causes. (b) The survival function of T is denoted by t p ( τ ) x and t p ( τ ) x = 1 - t q ( τ ) x = Pr { T > t } = Z t f T ( s ) ds, which is the probability that life ( x ) is still in the group at age x + t . (c) The total force of decrement is denoted by μ ( τ ) x ( t ) and μ ( τ ) x ( t ) = f T ( t ) 1 - F T ( t ) = d dt t q ( τ ) x t p ( τ ) x .
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