Reliability Theory — DU B.A. (Prog.) with Statistics as Major Discipline Previous Year Question Papers

DSE: Reliability Theory (NEP) Attempt five questions in all. (a) Show that for an exponential distributi.n with mean G, hazard rate is constant. (b) Consider a failure model with failure time of a certain component having a Weibull distribution with shape Parameter ^=0.5 and scale parameter y=2. Show that the hazard function is of decreasing function. (c) Obtain the reliability function of a system having two components when (i) units are operating in parallel, (ii) units are operating in series. The probability of success for each component is 0.6. (d) Suppose a certain system works normally at 2 out-of-4 systems but due to some requirement the system has t.o wol.k as 3-out-of-4 systems. Will there be a change in the reliability of the system? If yes, obtain the difference for p=0.9. (e) Consider a four-component system whose components are i.i.d. with constant failure rate ^i, i=1, 2 ,..., 4. If R(100) = 0.95 is the specifled reliability, find the individual component mean time to failure. ® State the difference between mean time to failure and mean time betw`een failures. Consider the pdf I(t)= (g) with t measured in` hours. Obtain the.reliability for a 100 hours operating life. The reliability function of a component is given by: Detemine the c-orresponding hazard rate function. (6x 3 - 18} 2. (a) Let the failure time T follow a weibull distrit)ution with density; -tp f ct)=%tp-te+, 8 >0,p>O,t>O where p is the shape parameter and Q is the scale parameter. Obtain the maximum likelihood estimates of 0 and p, under failure censored sampling scheme where the experiment is terminated after r failures and both G and p are unknown. (b) Explain bath-tub shaped hazard rate model and obtain the reliability function for the model. ' (12, 6) 4. (a) Let the failure time T follow a two-parameter Gamma distribution with shape parameter ^ and scale paraneter v having the density function Obtain. .the. hazard rate function, ieli`ability function and the mean time to failure of the system. (b) Suppose that the fail.ure time distribution follows a Weibull distribution with scale parameter ^ =1 and shape parameter v=0.5, find the mean, variance and the coefficient of variation of the failure time. (12, 6) (a) Obtain the'reliability function for the systems with (i) components in series, (ii) components in parallel. 5. {b) Obtain the reliabilit.v function foi. (i) 2-out-of-3 syste-in; (ii) simple 3unit parallel system where the probability of success for each component is 0.9. (12, 6) (a) Let the fafiure time T follow an exponential distribution with mean G. then obtain the UMVUE of G and the hazard rate function under time censored sampling scheme. {b) Obtain the reliability of the model where the strength.X and Stress Y follow independent exponential distrib`ution with mean i and i respectively! (12, 6) 6. Write short notes on any four of the following: (a) Components and system (b) Failure censored samples (c) k-out-of-in systems