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Unformatted text preview: Spring 2010 Pstat160b Handout #4 Residual lifetime and hazard rate Let X be a non-negative continuous random variable. It will be interpreted as a random time (lifetime for instance), with pdf f , cdf F , andtail probability orcomplementary cdf or survival probability F ( t ) = 1- F ( t ) = P ( X > t ), and mean or expected value . The residual lifetime distribution is defined by the conditional density of X | X > t , where t is fixed, that is, we define: t = current time (alive now) y = time of event in the future, starting at beginning, (actual time of death) = x + t x = residual time (how long before death) = y- t f r t ( x ) = f ( x + t ) 1- F ( t ) = f ( x + t ) F ( t ) = f ( y ) F ( t ) , x 0 residual time distribution, from current time (1) Note: Express in residual time x from current time t , with notation . r , or in absolute time y for original functions. Indeed, the above formula makes sense since we can calculate the probability F r ( s ) that the residual life does not exceed an extra s unit of time, F r ( s ) = P ( X t + s | X > t ) = P ( t < X t + s ) P ( X > t ) (2) = R t + s t f ( y ) dy 1- F ( t ) = R s f ( x + t ) dx F ( t ) = R s f r ( x ) dx F ( t ) (3) This means that the density is given by (1). The hazard rate (or HR for short), sometimes also referred to as force of mortality in actuarial science, is defined to be r ( t ) = f r t (0), that is r ( t ) = f ( t ) 1- F ( t ) = f ( t ) F ( t ) , t (4) r ( t ) is interpreted as the rate at which the event (death) occur at time t , or more precisely, P ( X [ t,t + h ] | X > t ) = F r ( h ) h r ( t ) for small h > To see this, the first equality is just the definition in (2), then replace the integral by the rectangle...
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This note was uploaded on 01/10/2011 for the course STAT PStat 160b taught by Professor Bonnet during the Spring '10 term at UCSB.
- Spring '10