EE255/CPS226 Continuous Random Variables

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EE255/CPS226Continuous Random Variables

• Dept. of Electrical Computer engineering
• Duke University
• Email bbm_at_ee.duke.edu, kst_at_ee.duke.edu

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Definitions

• Distribution function
• If FX(x) is a continuous function of x, then X is
  a continuous random variable.
• FX(x) discrete in x ? Discrete rvs
• FX(x) piecewise continuous ? Mixed rvs

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Probability Density Function (pdf)

• X continuous rv, then,
• pdf properties

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Exponential Distribution

• Arises commonly in reliability queuing theory.
• It exhibits memory-less (Markov) property.
• Related to Poisson distribution
• Inter-arrival time between two IP packets (or
  voice calls)
• Time interval between failures, etc.
• Mathematically,

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Exp Distribution Memory-less Property

• A light bulb is replaced only after it has
  failed.
• Conversely, a critical space shuttle component is
  replaced after some fixed no. of hours of use.
  Thus exhibiting memory property.
• Wait time in a queue at the check-in counter?
• Exp( ) distribution exhibits the useful
  memory-less property, i.e. the future occurrence
  of random event (following Exp( ) distribution)
  is independent of when it occurred last.

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Memory-less Property (contd.)

• Assuming rv X follows Exp( ) distribution,
• Memory-less property find P( ) at a future
  point.
• X gt u, is the life time, y is the residual life
  time

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Memory-less Property (contd.)

• Memory-less property
• If the components life time is exponentially
  distributed, then,
• The remaining life time does not depend on how
  long it has already working.
• If inter-arrival times (between calls) are
  exponentially distributed, then, time we need
  still wait for a new arrival is independent of
  how long we have already waited.
• Memory-less property a.k.a Markov property
• Converse is also true, i.e. if X satisfies Markov
  property, then it must follow Exp() distribution.

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Reliability Failure Rate Theory

• Reliability R(t) failure occurs after time t.
  Let X be the life time of a component subject to
  failures.
• N0 total components (fixed) Ns survived ones
• f(t)?t unconditional prob(fail) in the interval
  (t, t?t
• conditional failure prob.?

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Reliability Failure Rate Theory (contd.)

• Instantaneous failure rate h(t) (failures/10k
  hrs)
• Let f(t) (failure density fn) be EXP( ?). Then,
• Using simple calculus,

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Failure Behaviors

• There are other failure density functions that
  can be used to model DFR, IFR (or mixed) failure
  behavior

DFR
IFR
CFR
Failure rate
Time

• DFR phase Initial design, constant bug fixes
• CFR phase Normal operational phase
• IFR phase Aging behavior

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HypoExponential

• HypoExp multiple Exp stages.
• 2-stage HypoExp denoted as HYPO(?1, ?2). The
  density, distribution and hazard rate function
  are
• HypoExp results in IFR 0 ? min(?1, ?2)

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Erlang Distribution

• Special case of HypoExp All r stages are
  identical.
• X gt t Nt lt r (Nt no. of stresses applied
  in (0,t and Nt is Possion (param ?t). This
  interpretation gives,

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Gamma Distribution

• Gamma density function is,
• Gamma distribution can capture all three failure
  models, viz. DFR, CFR and IFR.
• a 1 CFR
• a lt1 DFR
• a gt1 IFR

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HyperExponential Distribution

• Hypo or Erlang ? Sequential Exp( ) stages.
• Parallel Exp( ) stages ? HyperExponential.
• Sum of k Exp( ) also gives k-stage HyperExp
• CPU service time may be modeled as HyperExp

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Weibull Distribution

• Frequently used to model fatigue failure, ball
  bearing failure etc. (very long tails)
• Weibull distribution is also capable of modeling
  DFR (a lt 1), CFR (a 1) and IFR (a gt1).
• a is called the shape parameter.

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Log-logistic Distribution

• Log-logistic can model DFR, CFR and IFR failure
  models simultaneously, unlike previous ones.
• For, ? gt 1, the failure rate first increases with
  t (IFR) after momentarily leveling off (CFR), it
  decreases (DFR) with time, all within the same
  distribution.

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Gaussian (Normal) Distribution

• Bell shaped intuitively pleasing!
• Central Limit Theorem mean of a large number of
  mutually independent rvs (having arbitrary
  distributions) starts following Normal
  distribution as n ?
• µ mean, s std. deviation, s2 variance (N(µ,
  s2))
• µ and s completely describe the statistics. This
  is significant in statistical estimation/signal
  processing/communication theory etc.

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Normal Distribution (contd.)

• N(0,1) is called normalized Guassian.
• N(0,1) is symmetric i.e.
• f(x)f(-x)
• F(z) 1-F(z).
• Failure rate h(t) follows IFR behavior.
• Hence, N( ) is suitable for modeling long-term
  wear or aging related failure phenomena.

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Uniform Distribution

• U(a,b) ? constant over the (a,b) interval

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Defective Distributions

• If
• Example

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Functions of Random Variables

• Often, rvs need to be transformed/operated upon.
• Y F (X) so, what is the distribution of Y ?
• Example Y X2
• If fX(x) is N(0,1), then,
• Above fY(y) is also known as the ?2 distribution
  (with 1-d of freedom).

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Functions of R Vs (contd.)

• If X is uniformly distributed, then, Y
  -?-1ln(1-X) follows Exp( ) distribution
• transformations may be used to synthetically
  generate random numbers with desired
  distributions.
• Computer Random No. generators may employ this
  method.

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Functions of R Vs (contd.)

• Given,
• A monotone differentiable function,
• Above method suggests a way to get the desired
  CDF, given some other simple type of CDF. This
  allows generation of random variables with
  desired distribution.
• Choose F to be F.
• Since, YF(X), FY(y) y and Y is U(0,1).
• To generate a random variable with X having
  desired distribution, choose generate U(0,1)
  random variable Y, then transform y to x F-1(y)
  .

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Jointly Distributed RVs

• Joint Distribution Function
• Independent rvs iff the following holds

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Joint Distribution Properties

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Joint Distribution Properties (contd)

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Order Statistics (min, max function)

• Define Yk ( known as the kth order statistics)
• Y1 minX1, X2, , Xn
• Yn maxX1, X2, , Xn
• Permute Xi so that Yi are sorted (ascending
  order)
• Y1 life of a system with series of
  components.
• Yn with parallel (redundant) set of
  components.
• Distribution of Yk ?
• Prob. that exactly j of Xi values are in (-8,y
  and remaining (n-j) values in (y, 8 is

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Sorted random sequence Yk
Observe that there are at least k Xis that
arelt y. Some of the remaining Xis may Also be
lt y
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Sorted RVs (contd)

• Using FY(y), reliability may be computed as,
• In general,

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Sorted RVs min case (contd)

• ith components life time EXP(?i), then,
• Hence, life time for such a system also has EXP()
  distribution with,
• For the parallel case, the resulting distribution
  is not EXP( )

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Sum of Random Variables

• Z F(X, Y) ? ((X, Y) may not be independent)
• For the special case, Z X Y
• The resulting pdf is,
• Convolution integral

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Sum of Random Variables (contd.)

• X1, X2, .., Xk are iid rvc, and Xi EXP(?),
  then rv (X1 X2 ..Xk) is k-stage Erlang with
  param ?.
• If Xi EXP(?i), then, rv (X1 X2 ..Xk) is
  k-stage HypoExp( ) distribution. Specifically,
  for ZXY,
• In general,

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Sum of Normal Random Variables

• X1, X2, .., Xk are normal iid rvc, then, the
  rv Z (X1 X2 ..Xk) is also normal
  with,
• X1, X2, .., Xk are normal. Then,
  
• follows Gamma or the ?2 (with n-deg of
  freedom) distributions

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Sum of RVs Standby Redundancy

• Two independent components, X and Y
• Series system (Zmin(X,Y))
• Parallel System (Zmax(X,Y))
• Cold standby the life time ZXY
• If X and Y are EXP(?), then,
• i.e., Z is Gamma distributed, and,
• May be extended 12 cold-standbys ? TMR

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k-out of-n Order Statistics

• Order statistics Yn-k1 of (X1, X2, .. Xn) is
• P(Yn-k1 ) HYPO(n? ,(n-1)? , k? )
• Proof by induction
• n2 case k2 ? Y1 parallel Y2 series
• F(Y1 ) (Fy)2 or F(Yn ) (Fy)n
• Y1 distribution? Y1 is the residual life time.
• If all Xi s are EXP(?) ? memory-less property
  I.e. residual life time is independent of how
  long the component has already survived.
• Hence, Y1 distribution is also EXP(?).

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k-out of-n Order Statistics (contd)

• Assume n-parallel components. Then, Y1 1st
  component failure or minX1, X2, .. Xn.
• 2nd failure would occur within Y2 Y1 minX1,
  X2, .. Xn. Xis are the residual times of
  surving components. But due to memory-less
  property, Xis are independent of past failure
  behavior. Therefore, F( minX2, X3, .. Xn) is
  EXP((n-1) ?). In general, for k-out of-n (k are
  working)
• Yn-k1 HYPO(n?, (n-1)?, .., k?)

EXP(n?)
EXP((n-1)?)
EXP((n-k1)?)
EXP(?)
Y1
Y2
Yn
Yn-k1