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Unformatted text preview: 4.12 The Hazard Function 151 Recall that in Chapter 1 we defined the reliability function R ( t ) of a compo nent as the probability that the component has not failed by time t . Thus, we can relate the reliability function R X ( x ) of X to its CDF as follows: R X ( x ) = P [ X > x ] = 1 P [ X x ] = 1 F X ( x ) Thus, the hazard function can be defined in terms of the reliability function as follows: h X ( x ) = f X ( x ) R X ( x ) We now show that by specifying the hazard function, we uniquely specify the reliability function and, hence, the CDF of a random variable. Since F X ( x ) = 1 R X ( x ) , we have that f X ( x ) = d dx F X ( x ) = d dx { 1 R X ( x ) } =  d dx R X ( x ) Thus, h X ( x ) = d dx F X ( x ) R X ( x ) = d dx R X ( x ) R X ( x ) =  d dx ln R X ( x ) Integrating both sides from 0 to x we obtain [ ln R X ( t ) ] x =  integraldisplay x h X ( t ) dt Since R X ( ) = 1, we have that ln R X ( x ) =  integraldisplay x h X ( t ) dt R X ( x ) = 1 F X ( x ) = exp braceleftbigg integraldisplay x h X ( t ) dt bracerightbigg Example 4.26 The time until a component fails is exponentially distributed with a mean of 200 hours. Given that the component has not failed after operating for 150 hours, calculate the hazard function. Solution Let T denote the time until the component fails. Then the PDF and CDF of T are given by f T ( t ) = e t , = 1 / 200 , t F T ( t ) = integraldisplay t f T ( x ) dx = 1 e t 152 Chapter 4 Special Probability Distributions Thus, the hazard function is given by h T ( t ) = f T ( t ) 1 F T ( t ) = e t e t = Since is the failure rate, we see that the hazard function in this case is a constant that is equal to the failure rate. trianglesolid Example 4.27 Determine the hazard function of a component whose time to failure X is the socalled Weibull random variable with parameters and and whose PDF and CDF are given by f X ( x ) = x  1 e x , x ; , F X ( x ) = 1 e x Solution The Weibull distribution is widely used in reliability modeling. When = 1, we obtain the exponential distribution; and when = 2, we obtain the Rayleigh distribution that is popularly used to model different types of interfer ence in communication systems. The hazard function of the Weibull distribution is given by h X ( x ) = f X ( x ) 1 F X ( x ) = x  1 e x e x = x  1 trianglesolid Example 4.28 The hazard function of a certain random variable Y is given by h Y ( y ) = . 5 y , y 0. What is the PDF of Y ? Solution The reliability function of Y is given by R Y ( y ) = exp braceleftbigg integraldisplay y h Y ( t ) dt bracerightbigg = exp braceleftbigg integraldisplay y . 5 tdt bracerightbigg = e . 25 y 2 = 1 F Y ( y ) Therefore, the CDF of Y is given by F Y ( y ) = 1 R Y ( y ) = 1 e . 25 y 2 Finally, the PDF of Y is given by f Y ( y ) = d dy F y ( y ) = . 5 ye . 25 y 2 y trianglesolid 4.13 Chapter Summary 153...
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This note was uploaded on 01/05/2010 for the course STAT 350 taught by Professor Carlton during the Fall '07 term at Cal Poly.
 Fall '07
 Carlton
 Probability

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