chap3f - Probability and Statistics with Reliability,...

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Copyright © 2005 by K.S. Trivedi 1 Probability and Statistics with Reliability, Queuing and Computer Science Applications: Second edition by K.S. Trivedi Publisher-John Wiley & Sons Chapter 3: Continuous Random Variables Dept. of Electrical & Computer Engineering Duke University Email: kst@ee.duke.edu URL: www.ee.duke.edu/~kst
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Copyright © 2005 by K.S. Trivedi 2 Definitions ± Distribution function: ± If is a continuous function of x, then X is a continuous random variable. ± : grows only by jumps Æ Discrete rv ± : both jumps and continuous growth Æ Mixed rv ) ( x F X ) ( x F X ) ( x F X
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Copyright © 2005 by K.S. Trivedi 3 Definitions (Contd.) Equivalence: ± CDF (Cumulative Distribution Function) ± Probability Distribution Function (PDF) but avoid this name as it can be confused with pdf (prob. density function) ± Distribution function ± or F X (t) orF(t) ) ( x F X
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Copyright © 2005 by K.S. Trivedi 4 probability density function (pdf) ± X: continuous rv, then, ± CDF and pdf can be derived from each other ± pdf properties:
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Copyright © 2005 by K.S. Trivedi 5 Definitions (Continued) ± Equivalence: pdf ± probability density function ± density function ± density ± f(t) = dt dF , ) ( ) ( ) ( 0 = = t t dx x f dx x f t F for a non-negative random variable
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Copyright © 2005 by K.S. Trivedi 6 Example 3.1 ± Random variable X : time (years) to complete a project ± f X clearly satisfies property (f1) . ± To be a pdf , it must also satisfy (f2) , ± Prob. of completing project in less than 4 months,
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Copyright © 2005 by K.S. Trivedi 7 Exponential Distribution ± Arises commonly in reliability & queuing theory. ± A non-negative continuous random variable. ± It exhibits memoryless property. ± Related to (discrete) Poisson distribution ± Often used to model ± Interarrival times between two IP packets (or voice calls) ± Service time distribution ± Time to failure, time to repair etc.
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Copyright © 2005 by K.S. Trivedi 8 Exponential Distribution ± The use of exponential distribution is an assumption that needs to be validated based on experimental data; if the data does not support the assumption, other distributions may be used ± For instance, Weibull distribution is often used to model time to failure; Markov modulated Poisson process is used to model arrival of IP packets
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Copyright © 2005 by K.S. Trivedi 9 Exponential Distribution ± Mathematically (CDF and pdf are given as): ± Also
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Copyright © 2005 by K.S. Trivedi 10 CDF of exponentially distributed random variable with λ = 0.0001 12500 25000 37500 50000
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Copyright © 2005 by K.S. Trivedi 11 Exponential Density Function (pdf)
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Copyright © 2005 by K.S. Trivedi 12 Memoryless property ± Assume X > t, i.e., We have observed that the component has not failed until time t. ± Let Y = X - t , the remaining (residual) lifetime y Y e t X P t y X t P t X t y X P t X y Y P t y G λ = > + < = > + = > = 1 ) ( ) ( ) | ( ) | ( ) | (
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Copyright © 2005 by K.S. Trivedi 13 Memoryless property ± Thus G Y (y|t)is independent of tand is identical to the original exponential distribution of X.
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chap3f - Probability and Statistics with Reliability,...

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