Conditional Probability: Theory and Applications

1. Probability and Statistics with Reliability, Queuing and Computer Science Applications: Chapter 5 on Conditional Probability and Expectation Dept. of Electrical & Computer engineering Duke University Email: bbm@ee.duke.edu, kst@ee.duke.edu

2. Conditional pmf • Conditional probability: • Above works if x is a discrete rv. • For discrete rv’s X and Y, conditional pmf is, • Above relationship also implies, • Hence we have another version of the theorem of total probability Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

3. Independence, Conditional Distribution • Conditional distribution function • Using conditional pmf, Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

4. Example • Two servers p: prob. that the next job goes to server A k jobs A p Poisson ( λ) Job stream Bernoulli trial n jobs 1-p B • n total jobs, k are passed on to server A Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

5. Conditional pdf • For continuous rv’s X and Y, conditional pdf is, • Also, • Independent X, Y  • Marginal pdf (cont. version of the TTP), • Conditional distribution function Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

6. Conditional Reliability • Software system after having incurred (i-1) faults, • Ri(t) = P(Ti > t) (Ti : inter-failure times) • Ti : independent exponentially distributed Exp(λi). • λi : Failure rate itself may be random, then • Conditional reliability: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

7. Mixture Distribution • Conditional distribution: continuous and discrete rvs combined. • Examples: (Response time | that there k processors), (Reliability| k components) etc. (Y: continuous, X:discrete) • Compute server with r classes of jobs (i=1,2,..,r) • Hence, Y follows an r-stage HyperExpo distribution. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

8. Mixture Distribution (contd.) • What if fY|X(y|i) is not Exponential? • The unconditional pdf and CDF are, • Using LST, • Moments can now be found as, Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

9. Mixture Distribution (contd.) • Such Mixture distrib.: arise in reliability studies. • Software system: Modules (or objects) may have been written by different groups or companies, ith group contributes ai fraction of modules and has reliability characteristic given by Fi. • Gp#1: EXP( λ1) (αfrac); Gp#2: r-stage Erlang (1- α frac) Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

10. Mixture Distribution (contd.) • Y:continuous; X: continuous or uncountable, e.g., life time Y depends on the impurities X. • Finally, Y:discrete; X: continuous Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

11. Mixture Distribution (contd.) • X: web server response time; Y: # of requests arriving while a request being serviced. • For a given value of X=x, Y is Poisson, • The joint pdf is, f(x,y) = pY|X(y|x)fX(x) • Unconditional pmf pY(y) = P(Y=y) • With (λ+μ)x = w, Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

12. Conditional Moments • Conditional Expectation is E[Y|X=x] or E[Y|x] • E[Y|x]: a.k.a regression function • For the discrete case, • In general, then, Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

13. Conditional Moments (contd.) • This can be specialized to: • kth moment of Y: E[Yk|X=x] • Conditional MGF, MY|X(θ |x) = E[eθY|X=x] • Conditional LST, LY|X(s|x) = E[e-sY|X=x] • Conditional PGF, GY|X(z|x) = E[zY|X=x] • Total Expectation: • Total moments: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

14. Conditional Moments (contd.) • Total transforms: • In the previous example, • Total expectation: • Therefore, we can also talk of conditional MTTF • MTTF may depend on impurities or operating temp. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

15. Conditional MTTF • Y: time-to-failure may depend on the temperature, and the conditional MTTF may be: • Let Temp be normal, • Unconditional MTTF is: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

16. Imperfect Fault Coverage • Hybrid k-out of-n system, with m cold standbys. • Reliability depends on recovery from a failure. What if the failed module cannot be substituted by a standby? These are called not covered faults. • Probability that a fault is covered is c (coverage factor) Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

17. Fault Handling Phases • Fault handling involves 3-distinct phases. • Finite success probability for each phase  finite coverage. c = P(“ok recovery”|”fault occurs”) = P(“fault detected” & “fault located” & “fault corrected” | “fault occurs”) = cd.cl.cr Fault Processing Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

18. Near Coincident Faults • Coincident fault: 2nd fault occurs while the 1st one has not been completely processed. • Y: Random time to process a fault. • X: Time at which coincident fault occurs (EXP(γ)). • Fault coverage: prob. that Y < X Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

19. Near Coincidence: Fault Coverage • Fault handling has multiple phases. This gives: • X:Life time of a system with one active + one standby • λ: Active component’s failure rate; • Y = 1  fault covered; Y = 0  fault not covered. • c=0 or c=1? Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

20. Life Time Distribution-Limited Coverage • fX|Y(t|0): life time of the active comp. ~EXP(λ) • fX|Y(t|1): life time of active+standby 2-stage Erlang • Joint density fn: • Marginal density fn: • Reliability Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University