Part-1-Notes-232-2010W

Part-1-Notes-232-2010W - Summary of Lecture Notes - ACTSC...

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Unformatted text preview: Summary of Lecture Notes - ACTSC 232, Winter 2010 Part 1 - Survival Distributions and Life Tables 1.1 Future Lifetime Random Variables (a) Let ( x ) denote a life or person aged x ( 0) at time 0 or at the current time. (b) Let T x denote the time-until-death of ( x ). That is to say that ( x ) will die at age x + T x or will die in T x years or T x is the death time of ( x ). The random variable T x is called the future lifetime of a life ( x ). Note that T x > 0 is a continuous random variable. (c) The distribution function (d.f.) of T x is denoted by F x ( t ) = Pr { T x t } , t . The survival distribution (s.f.) of T x is denoted by S x ( t ) = Pr { T x > t } = 1- F x ( t ) , t . The probability density function (p.d.f.) of T x is denoted by f x ( t ) = d dx F x ( t ) =- d dx S x ( t ) , t . (d) Let T denote the age at the death of a newborn life or the future lifetime of (0), or the age at the death of ( x ) or the future lifetime of ( x ) from his birth. Note that T > 0 is a continuous random variable. When we view T as the age at the death of ( x ) or the future lifetime of ( x ) from his birth, then T = x + T x and T x is as a conditional random variable conditioning on T > x . In this course, we explain T x in this way. (e) The distribution function (d.f.) of T is denoted by F ( x ) = Pr { T x } . The survival distribution (s.f.) of T is denoted by S ( x ) = 1- F ( x ) = Pr { T > x } . The probability density function (p.d.f.) of X is denoted by f ( x ) = d dx F ( x ) =- d dx S ( x ) . 1 (f) Relationships between the distribution of T x and the distribution of T : Pr { T x t } = Pr { T x + t | T > x } , Pr { T x > t } = Pr { T > x + t | T > x } . That means F x ( t ) = F ( x + t )- F ( x ) 1- F ( x ) = S ( x )- S ( x + t ) S ( x ) S x ( t ) = S ( x + t ) S ( x ) . (g) Review of the properties of distribution and survival functions: Let the df and sf of a r.v. Y be F ( x ) = Pr { Y x } and S ( x ) = Pr { Y > x } . i. 0 F ( x ) 1 (0 S ( x ) 1). ii. F ( x ) ( S ( x )) is a non-decreasing (non-increasing) and right-continuous function. iii. lim t F ( x ) = 1 and lim t - F ( x ) = 0 (lim t S ( x ) = 0 and lim t - S ( x ) = 1). iv. For any a < b , Pr { a < Y b } = F ( b )- F ( a ) = S ( a )- S ( b ) . v. If F ( x ) is continuous, then for any a < b , Pr { a < Y b } = Pr { a < Y < b } = Pr { a Y b } = Pr { a Y b } = F ( b )- F ( a ) = S ( a )- S ( b ) . vi. If Y is limited or bounded from above, say Y , then F ( x ) = 1 , x S ( x ) = 0 , x . Note that if Y represents future lifetime, the is called the limiting age. vii. The conditions (i)-(iii) are sufficient and necessary for a function to be a distri- bution (survival) function of a random variable....
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Part-1-Notes-232-2010W - Summary of Lecture Notes - ACTSC...

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