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ACTSC232-Ch2 - Chapter 2 Survival models Chapter 2 Survival...

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Chapter 2 Survival models Chapter 2 Survival models ACTSC 232 Introduction to Actuarial Mathematics Tianxiang Shi Department of Statistics and Actuarial Science University of Waterloo Winter 2012 Tianxiang Shi([email protected])
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Chapter 2 Survival models Outline 1 Brief review of Prob. & Stat 2 The future lifetime random variable 3 The force of mortality 4 Moments of future lifetime T x 5 Curtate future lifetime K x Tianxiang Shi([email protected])
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Chapter 2 Survival models Brief review of Prob. & Stat Random variable Definition A random variable (r.v.) is a mapping from a set of random events to the real line R . discrete r.v. take values in a countable set (e.g. { 0, 1 } ). continuous r.v. take values in some intervals of real numbers (e.g. [0, 1]). Tianxiang Shi([email protected])
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Chapter 2 Survival models Brief review of Prob. & Stat Cumulative distribution function Definition A cumulative distribution function (c.d.f.) of a r.v. X is defined as F ( x ) Pr( X x ) for x R . A function F ( x ) to be a proper c.d.f. iff : Tianxiang Shi([email protected])
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Chapter 2 Survival models Brief review of Prob. & Stat Cumulative distribution function Definition A cumulative distribution function (c.d.f.) of a r.v. X is defined as F ( x ) Pr( X x ) for x R . A function F ( x ) to be a proper c.d.f. iff : F ( x ) is non-decreasing as a function of x Tianxiang Shi([email protected])
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Chapter 2 Survival models Brief review of Prob. & Stat Cumulative distribution function Definition A cumulative distribution function (c.d.f.) of a r.v. X is defined as F ( x ) Pr( X x ) for x R . A function F ( x ) to be a proper c.d.f. iff : F ( x ) is non-decreasing as a function of x lim x →-∞ F ( x ) = 0 Tianxiang Shi([email protected])
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Chapter 2 Survival models Brief review of Prob. & Stat Cumulative distribution function Definition A cumulative distribution function (c.d.f.) of a r.v. X is defined as F ( x ) Pr( X x ) for x R . A function F ( x ) to be a proper c.d.f. iff : F ( x ) is non-decreasing as a function of x lim x →-∞ F ( x ) = 0 lim x →∞ F ( x ) = 1 Tianxiang Shi([email protected])
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Chapter 2 Survival models Brief review of Prob. & Stat Survival function Definition A survival function (s.f.) of a r.v. X is defined as S ( x ) Pr( X > x ) for x R . A function S ( x ) to be a proper s.f. iff : Tianxiang Shi([email protected])
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Chapter 2 Survival models Brief review of Prob. & Stat Survival function Definition A survival function (s.f.) of a r.v. X is defined as S ( x ) Pr( X > x ) for x R . A function S ( x ) to be a proper s.f. iff : S ( x ) is non-increasing as a function of x lim x →-∞ S ( x ) = 1 lim x →∞ S ( x ) = 0 Relationship : S ( x ) = 1 - F ( x ) Tianxiang Shi([email protected])
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Chapter 2 Survival models Brief review of Prob. & Stat Probability density/mass function Probability density function (p.d.f.) for a continuous r.v. X : f ( x ) = d dx F ( x ) for x R F ( b ) = R b -∞ f ( x ) dx Pr( a < X b ) = Pr( a X < b ) = R b a f ( x ) dx Probability mass function (p.m.f.) for a discrete r.v. X : f ( k ) = Pr( X = k ) for possible value k of X F ( b ) = k b f ( k ) Pr( X b ) = Pr( X < b ) is generally NOT TRUE !
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  • Winter '08
  • MATTHEWTILL
  • Probability theory, probability density function, Cumulative distribution function, survival models, Tianxiang Shi

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