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Probability

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					Probability and Reliability
Outline:
                                        How do I describe a
                                         random variable?

    Basic descriptive statistics
    Basic probability concepts
    Probability distributions
                                        How do I calculate
    Binomial distributions               probabilities?
    Exponential distributions
                                    How do I calculate
    Reliability                        reliability

    Confidence intervals
Basic descriptive statistics
     Mean                  Average (expected) value

     Median            50th-percentile value
                        (50% above, 50% below)
     Mode                       Most likely value

     Standard deviation         Measure of dispersion
                                 about the mean
     Coefficient of variation      Dimensionless measure
                                    of variability (COV=s/m)
     Higher-order moments
                                     Skewness, flatness, etc.
Basic probability concepts
                                Three (3) basic axioms
    Axioms of probability

    Discrete and continuous distributions

    PDF’s and CDF’s

    Probability calculations
                                “What is the probability of …?”
Statistical independence
(important concept)



         A                     B




                            IF A and B are independent
 A “intersects”B

        P(A  B) = P(A)·P(B/A) = P(A) P(B)
Sample probability functions
(density, and cumulative distribution functions)




                                                                    a
                                          F X (a )  P[ X  a ]     f X ( x)dx
                                                                    
Some common distributions

 Discrete                  Continuous
      Binomial                Uniform
      Pascal (r)              Normal
      Geometric (r=1)         Lognormal
      Poisson                 Gamma (r)
      Negative Binomial       Exponential (r=1)
Probability calculations
    What is the probability of a random variable being
     equal to some value ?

    What is the probability of being less than some value?
                                               a
                    F X (a )  P[ X  a ]      f X ( x)dx
                                               


    What is the probability of being between two values?
                              b
             P[a  X  b]     f X ( x)dx     FX (b)  FX (a)
                              a
Binomial distribution
 Binomial distribution is a discrete distribution, and is used for situations
 where individual trials, experiments, or tests have only two possible
 outcomes such as success/failure, go/no-go, defectetive/non-defective or
 pass/fail.

                       N!
p( r / N , p )                 p (1  p ) N  r
                                 r

                   r!( N  r )!


N is the sample size ( in a trial)
r is the number of events of interest
observed in the sample (i.e. failures, or
defectives)
p is the proportion of events of interest
in the population
p(r / N, p) also can be written as
P (r, N, p) not to be confused with p.
Binomial Distribution (cont’d)
Example 1:
  The proportion of defectives in a product line is 1%. Take a random
  sample of 25. What is the probability that this sample has one
  defective ?
  N = 25, p = 0.01, r = 1
                                            N!
        p(r  1 / N  25, p  0.01)                 p (1  p ) N r  .1964
                                                      r

                                        r!( N  r )!

   What if p = 0.001 ?
                                             N!
         p(r  1 / N  25, p  0.001 )               p (1  p ) N r  .0244
                                                       r

                                         r!( N  r )!
Binomial Distribution (cont’d)
Example 2:
  The failure rate of a device on demand is 10-4. What is the probability
  that 3 out of 4 devices fail at demand ?
  N = 4, r = 3, p = 10-4

                                       N!
p(r  3 / N  4, p  10  4 )                 p (1  p ) N r  3.996  10 12 ~ Np
                                                r

                                  r!( N  r )!
        Exponential Distribution
The Exponential Distribution is widely used in reliability
engineering to estimate failure probability as function of
time.                                   F ( , T )  P(T  T /  )
                                                          0            0
                                                     T0

                       p' (T ,  )   exp(T )     p '(T ,  )dT 1  exp(T0 )
                                                     0
                          is the failure rate
                         T is time to fail
             0.1

            0.09

            0.08

            0.07

            0.06
   P(,T)




            0.05

            0.04

            0.03

            0.02

            0.01

              0
                   0
                                     T
Exponential Distribution
(cont’d)
Example 1:
  An electric motor's constant failure rate is 0.0004 failures/hr. Calculate
  probability of failure for a 150 hr mission.
  Operating time starts at t = 0.
   = 0.0004, T0 = 150 hr

             F ( , T0 )  P(T  T0 /  )  1  exp(T0 )  0.0582
  Mean life expectancy E(T) = 1/  = 2500 hr


  What if  = 0.004 ?

             F(,T0) = 0.4512,        E(T) = 1/  = 250 hr
   Exponential Distribution
   (cont’d)
Example 2:
  A CD player has an average record of successfully operating and
  providing listening enjoyment for more than 5,000 hours on the
  average before requiring repairs. A customer is planning on buying
  a CD player for installation in a boat that will be taking an extended
  cruise that will demand 4,000 hours of play before being able to
  obtain repairs or routine maintenance. How reliable will the CD
  player be for the customer? (Assume an exponential distribution)
Exponential Distribution
(cont’d)
Mean life expectancy E(T) = 1/  = 5000 hr, so  = 1/5000 = 0.00002
per hr



       F ( , T0  4000)  P(T  T0  4000 /  )  1  exp(T0 )  0.5507
       R(T0  4000 /  )  1  F ( , T0  4000)  0.4493


    Reliability R(t) is the probability that no system failure
    will occur in a given time interval.
    As expected this is not a good reliability for such a task.
Reliability as a function of System Complexity




    # of components    Component          Component
    in Series          Reliability =      Reliability =
                       99.999%            99.99%
    100                99.9               99.01
    250                99.75              97.53
    500                99.50              95.12
    1000               99.01              90.48
    10,000             90.48              36.79
    100,000            36.79              0.01
Confidence Intervals
Example 1:
  For quality control, a random sample of 25 are taken from a large
  population of products. What is the defective proportion, if one
  defective product is found in the sample ?
  N = 25, r = 1 p = ?



                      pe = r/N = 1/25 = 0.04
  Confidence Intervals (cont’d)
                           Z2   pe (1  pe ) Z 2
From Binomial:
                 N
                      pe   Z                                          
               N Z 2
                           2N        N        4N 2
Where pe is the observed p, and Z is the standard score for confidence asked

                 For a 95% confidence interval, Z = 1.96
                 pU = 0.1954
                 pL = 0.0071
   It means that if the same population is sampled on numerous occasions
   and interval estimates are made on each occasion, the resulting intervals
   would bracket the true population parameter in approximately 95% of the cases.

                   N              pU              pL
                   25             0.1954          0.0071
                   100            0.0984          0.0157
                   500            0.0610          0.0260
     Confidence Intervals (cont’d)
     Example 1:
        An electric motor fails after 150 hr operation. Estimate the 90% confidence
        interval of the failure rate .
                                             e= 1/150 = 0.0067 (failures per hour)
Use exact confidence interval for Poisson process:
re is the observed r=1 (one failure).  is for 100(1-) % confidence ; that is  = 0.1

                               re                               re
                                                                  e  U T (U T ) r
                              
                              r 0
                                            p(r / U , T )  
                                                             r 0          r!
                                                                                      /2

                                    re 1                    re 1
                                                                   e  LT (LT ) r
                                   r 0
                                             p(r / L , T )  
                                                              r 0         r!
                                                                                    1 / 2


                                                     U= 0.032, L= 0.000355
                                                           U <  < L
Questions / Discussion

				
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