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lect15 - HAZARD FUNCTIONS TRANSFORMATION OF VARIABLES...

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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
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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 0 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
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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
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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 0 λ ( s ) ds = . S ( t ) = 1 - F ( t ) = exp ( - R t 0 λ ( s ) ds ) f ( t ) = - d dt S ( t ) = λ ( t ) exp ( - R t 0 λ ( s ) ds ) Example: If T has the memoryless property with density f ( t ) , then λ ( t ) = for all t 0, so T .
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