##### Reference no: EM13671287

**Problem 1. **The Weibull hazard rate function λ(t) is given by:

λ(t) = (β/θ)t

Where β is the shape parameter and θ is the scale parameter. Obtain an expression for a reliability function.

**Problem 2.**

Assume load and capacity are exponentially and normally distributed, respectively. The mean load is equal to 14000psi and the values of the mean and standard deviation associated with the normal distribution are 21000 and 3000 psi respectively. Calculate the reliability.

**Problem 3. **

The following numbers of bends to failure were recorded for 20 paper clips:

11,29,15,20,19,11,12,9,9,8,13,20,11,22,20,9,25,19,11 and 10.

A. Make a nonparametric plot of R(t), the reliability.

B. Attempt to fit your data to Weibull and Normal Distributions then determine the parameters.

C. Briefly discuss your results.

**Problem 4.**

A device has a constant failure rate of 0.7/year.

A. What is the probability that the device will fail during the *second* year of operation.

B. If upon failure the device is immediately replaced, what is the probability that there will be more than one failure in 3 years operation?

**Problem 5. **

There exists a nuclear reactor plant with a system to deliver emergency cooling with 2 pumps and 4 valves. In the event of an accident the protection system delivers an actuation signal to the two identical pumps and the 4 identical valves. The pumps start up, the valves open and the liquid coolant is delivered to the reactor. The following failure probabilities are found to be significant:

P(ps)=10^{-5}, the probability that the protection system will not deliver a signal to the pump and the valve actuators.

P(p)= 2 x 10^{-2}, the probability that a pump will fail to start when the actuation signal is received.

P(v)= 10^{-1}, the probability that a valve will fail to open when the actuation signal is received.

P(r) =0.5 x 10^{-5}, the probability that the reservoir will be empty at the time of the accident.

A) Draw a fault tree for the failure of the system to deliver any coolant to the primary system in the event of an accident.

B) Evaluate the probability that such a failure will take place in the event of an accident.