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ch02 - Chapter 2 Survival models ACTSC 232 — Introduction...

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Unformatted text preview: Chapter 2. Survival models ACTSC 232 — Introduction to Actuarial Mathematics Department of Statistics and Actuarial Science University of Waterloo Fall 2011 Instructor: Chengguo Weng C. Weng ([email protected]) – p. 1/42 Review on Prob. & Stat • Random variable XBox e.g. toss a coin and let X = 1 , if “head" shows up; , otherwise. XBox Def: a r.v. is a mapping from a set of random events to the real line R . XBox Continuous r.v.’s RHD take values in some interval of real numbers, e.g., [0,1] XBox Discrete r.v.’s RHD take integer values, or more generally, values in a countable set. XBox In this course, we model the future lifetime of an individual as a continuous r.v. with values on interval [0 , ∞ ) or [0 ,ω ] for some limit age ω > . RHD Denote T x to be the future lifetime of an individual currently aged x RHD K x := ⌊ T x ⌋ is called curtate future lifetime of a person who ages x currently. K x is a discrete r.v.. C. Weng ([email protected]) – p. 2/42 Review on Prob. & Stat • Cumulative distribution function (c.d.f.) XBox Def: the c.d.f. of a r.v. X is defined by F X ( x ) := Pr( X ≤ x ) for x ∈ R XBox A function F ( x ) is a proper c.d.f. iff the following three conditions holds: RHD F ( x ) is non-decreasing as a function of x RHD lim x →−∞ F ( x ) = 0 RHD lim x →∞ F ( x ) = 1 • Survival function (s.f.) XBox Def: the s.f. of a r.v. X is defined by S X ( t ) = Pr( X > t ) for t ∈ R . XBox Necessary and sufficient conditions for a function S ( x ) to be a proper s.f.: RHD S ( x ) is non-increasing as a function of x RHD lim x →−∞ S ( x ) = 1 RHD lim x →∞ S ( x ) = 0 C. Weng ([email protected]) – p. 3/42 Review on Prob. & Stat • Survival function of T x XBox Def: S T x ( t ) = Pr( X > t ) for t ≥ . XBox We must have the following three results: RHD S T x ( t ) is non-increasing as a function of t ; RHD lim t → S T x ( t ) = 1 ; RHD lim t →∞ S T x ( t ) = 0 , or S T x ( ω ) = 0 if a limit age ω exists. C. Weng ([email protected]) – p. 4/42 Review on Prob. & Stat • Probability density function (p.d.f.) XBox defined only for continuous r.v.’s. XBox Def: f X ( t ) = d dt F X ( t ) for t ∈ R . RHD F X ( a ) = Pr( X ≤ a ) = varint a −∞ f X ( x ) dx for a ∈ R RHD Pr( a < X ≤ b ) = varint b a f X ( x ) dx , and Pr( a < X ≤ b ) = Pr( a ≤ X < b ) = Pr( a < X < b ) for all a,b ∈ R , a < b and a continuous r.v. X . • Probability mass function (p.m.f.) XBox defined only for discrete r.v.’s. XBox Def: f X ( k ) = Pr( X = k ) for possible value k of X . RHD F X ( a ) = Pr( X ≤ a ) = ∑ k ≤ a f X ( k ) for a discrete r.v. X . RHD Pr( a < X ≤ b ) = Pr( a ≤ X < b ) = Pr( a < X < b ) is NOT true generally....
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ch02 - Chapter 2 Survival models ACTSC 232 — Introduction...

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