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

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the 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 (c2weng@uwaterloo.ca) 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 , ] for some limit age > . 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 (c2weng@uwaterloo.ca) 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 (c2weng@uwaterloo.ca) 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 ( ) = 0 if a limit age exists. C. Weng (c2weng@uwaterloo.ca) 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

This note was uploaded on 02/08/2012 for the course ACTSC 232 taught by Professor Matthewtill during the Fall '08 term at Waterloo.

Page1 / 42

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online