lect15 - HAZARD FUNCTIONS TRANSFORMATION OF VARIABLES...

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Unformatted text preview: HAZARD FUNCTIONS TRANSFORMATION OF VARIABLES Outline HAZARD FUNCTIONS TRANSFORMATION OF VARIABLES CAUCHY DISTRIBUTIONS MANY-TO-ONE TRANSFORMATIONS 1 / 13 Xinghua Zheng Lect 15: Hazard functions; Transformation of variables HAZARD FUNCTIONS TRANSFORMATION OF VARIABLES EXPONENTIAL RANDOM VARIABLES • Suppose N ( t ) is a Poisson process with rate λ . Let T = T 1 be the waiting time for the first event. • Then T has F T ( t ) = P [ T ≤ t ] = P [ N ( t ) ≥ 1 ] = 1- exp (- λ t ) , f T ( t ) = F T ( t ) = λ exp (- λ t ) . • T is said to have an exponential distribution with (rate) parameter λ . 2 / 13 Xinghua Zheng Lect 15: Hazard functions; Transformation of variables HAZARD FUNCTIONS TRANSFORMATION OF VARIABLES HAZARD FUNCTIONS • Let T be a nonnegative continuous random variable with density f and cdf F . Think of T as the lifetime of some item. • The ratio of the chance that the item dies in the next dt time units, given that it has lived at least t time units to dt is λ ( t ) = P [ t < T ≤ t + dt | T > t ] dt = P [ t < T ≤ t + dt ] dt P [ T > t ] = f ( t ) dt dt ( 1- F ( t )) =- d dt ( log ( 1- F ( t ))) . • λ ( t ) is called the death rate , or failure rate , or hazard rate at time t . S ( t ) := P [ T > t ] = 1- F ( t ) is called the survival probability at time t . • Example: If T has an exponential distribution with parameter λ , then λ ( t ) = . 3 / 13 Xinghua Zheng Lect 15: Hazard functions; Transformation of variables HAZARD FUNCTIONS TRANSFORMATION OF VARIABLES HAZARD FUNCTIONS, ctd • In general, if you know the hazard function λ ( t ) of T , what can you say about the distribution of T ? • λ ( t ) =- d dt ( log ( 1- F ( t ))) ⇒ R t λ ( s ) ds = ....
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This note was uploaded on 11/27/2011 for the course ISOM 3540 taught by Professor Zheu during the Spring '11 term at HKUST.

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lect15 - HAZARD FUNCTIONS TRANSFORMATION OF VARIABLES...

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