This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 nondecreasing 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 nonincreasing 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 nonincreasing 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....
View
Full Document
 Fall '08
 MATTHEWTILL
 Actuarial Science, Probability theory, probability density function, Cumulative distribution function, C. Weng

Click to edit the document details