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					                            Stat-613 (Reliability)

                                   Sheet no. 1
1. Consider the following reliability function, where t is in hours:
          R(t) = 1 / (0.001t + 1), t  0,
   (a) Find the reliability after 100 operating hours; after 1000 operating hours.
   (b) Derive the hazard rate function. Is it an increasing or a decreasing failure rate?

2. A component has the following linear hazard rate, where t is in years:
          (t) = 0.4 t, t  0,
   (a) Find R(t) and determine the probability of a component failing within the first
   month of its operation.
   (b) What is the design life if a reliability of 0.95 is desired?

3. The time-to-failure density function (PDF) for a system is
            f(t) = 0.01, 0  t  100 days
   (a) R(t)
   (b) The hazard rate function
   (c) The MTTF
   (d) The standard deviation
   (e) The median time to failure

4. The failure distribution is defined by
           f(t) = 3t2 /109, 0  t  100 hr
   (a) What is the probability of failure within a 100-hr warranty period?
   (b) Compute the MTTF.
   (c) Find the design life for a reliability of 0.99.

5. For
          R(t )  e  0.001t , t  0
   (a) Compute the reliability for a 50-hr mission.
   (b) Show that the hazard rate is decreasing.

6. For the following PDF
           f(t) = 0.2 – 0.02 t, 0  t  10 yr
   (a) Show that the hazard rate function is increasing.
   (b) Find the MTTF.
   (c) Find the median time to failure.
   (d) Find the mode time to failure.
   (e) Compute the standard deviation.
   Dr. Hanan Page 1
7. For the following PDF
            f (t )             ,t0
                     ( t  10)3
    (a) Derive the reliability function and determine the reliability for the first year of
    (b) Compute the MTTF.
    (c) What is the design life for a reliability of 0.95?

8. A uniform failure distribution has the characteristic that equal intervals of time
   have equal failure probabilities. The density function is given by
           f (t )  , 0  t  b
   Analyze this general failure distribution by finding F(t), R(t), (t), MTTF, Tmed,
   Tmode and .

9. The reliability of a turbine blade can be expressed by the following:
            R( t )  (1  ) , 0  t  t0
   where t0 is the maximum life of the blade.
   (a) Show that the blades are experiencing wearout.
   (b) Compute the MTTF as a function of the maximum life.
   (c) If the maximum life is 2000 operating hours, what is the design life for a
       reliability of 0.9?

10. For a system having a linearly increasing hazard rate function, i.e.
             = at, a > 0, t  0,
    (a) Derive the reliability function.
    (b) Find the density function.
    (c) Find expressions for median time to failure and the mode.

11. A household appliance is advertised as having more than a 10-yr life. If the
    following is its PDF, determine its reliability for the next 10 yr if it has survived a
    1-yr warranty period:
             f ( t )  0.1(1  0.05 t )3 , t  0,
    What is its MTTF before the warranty period, and what is its MTTF after the
    warranty period assuming it has still survived?

12. Show that if the hazard rate function is decreasing, the PDF, f(t), is also a
    decreasing function and its mode must therefore occur at t=0.

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