Download Discrete Random Variables

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: [email protected], [email protected]
5/25/2017
1
Conditional pmf
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Conditional probability:
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Above works if x is a discrete rv.
For discrete rv’s X and Y, conditional pmf is,
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Above relationship also implies,
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Hence we have another version of the theorem of
total probability
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University
Independence, Conditional Distribution
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Conditional distribution function
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Using conditional pmf,
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University
Example
Two servers
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Poisson ( λ)
Job stream
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k jobs
Bernoulli trial
n jobs
p
A
1-p
B
p: prob. that the next job
goes to server A
n total jobs, k are passed on to server A
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University
Conditional pdf
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For continuous rv’s X and Y, conditional pdf is,
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Also,
Independent X, Y 
Marginal pdf (cont. version of the TTP),
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Conditional distribution function
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Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University
Conditional Reliability
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Software system after having incurred (i-1) faults,
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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
Mixture Distribution
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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)
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Hence, Y follows an r-stage HyperExpo distribution.
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Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University
Mixture Distribution (contd.)
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What if fY|X(y|i) is not Exponential?
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The unconditional pdf and CDF are,
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Using LST,
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Moments can now be found as,
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University
Mixture Distribution (contd.)
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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) (α frac); Gp#2: r-stage Erlang (1- α frac)
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University
Mixture Distribution (contd.)
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Y:continuous; X: continuous or uncountable, e.g.,
life time Y depends on the impurities X.
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Finally, Y:discrete; X: continuous
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University
Mixture Distribution (contd.)
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X: web server response time; Y: # of requests arriving
while a request being serviced.
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For a given value of X=x, Y is Poisson,
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The joint pdf is, f(x,y) = pY|X(y|x)fX(x)
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Unconditional pmf pY(y) = P(Y=y)
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With (λ+μ)x = w,
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University
Conditional Moments
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Conditional Expectation is E[Y|X=x] or E[Y|x]
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E[Y|x]: a.k.a regression function
For the discrete case,
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In general, then,
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University
Conditional Moments (contd.)
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This can be specialized to:
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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]
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Total Expectation:
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Total moments:
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University
Conditional Moments (contd.)
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Total transforms:
In the previous example,
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Total expectation:
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Therefore, we can also talk of conditional MTTF
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MTTF may depend on impurities or operating temp.
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University
Conditional MTTF
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Y: time-to-failure may depend on the temperature,
and the conditional MTTF may be:
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Let Temp be normal,
Unconditional MTTF is:
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Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University
Imperfect Fault Coverage
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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
Automatic fault diagnosis
Success
1-d
d
Automatic fault location
Failure
Manual diagnosis
Manual fault location
Success
a
1-a
Failure
Fault repair
Fault repair
Manual location
Fault repair
Fault Handling Phases
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Fault handling involves 3-distinct phases.
Fault Processing
cl
cd
Detect
Locate
1-cd
1-cl
cr
Recover
Coverage
Success
1-cr
Coverage
Failure
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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
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University
Near Coincident Faults
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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)).
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Fault coverage: prob. that Y < X
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Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University
Near Coincidence: Fault Coverage
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Fault handling has multiple phases. This gives:
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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.
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c=0 or c=1?
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Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University
Life Time Distribution-Limited Coverage
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fX|Y(t|0): life time of the active comp. ~EXP(λ)
fX|Y(t|1): life time of active+standby 2-stage Erlang
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Joint density fn:
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Marginal density fn:
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Reliability
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Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University