## Question

###### Consider system comprising three components_ The system requires at least two out of the three components to working for the system to work: Components fail independently and the time to failure for each component has an exponential distribution with mean of one year:Determine expression for the probability that the system is working at time Hence, give the probability density function for the time to failre of the system_ mark Determine the mean and variance of the time to failure for the syste

Consider system comprising three components_ The system requires at least two out of the three components to working for the system to work: Components fail independently and the time to failure for each component has an exponential distribution with mean of one year: Determine expression for the probability that the system is working at time Hence, give the probability density function for the time to failre of the system_ mark Determine the mean and variance of the time to failure for the system_ [2 marks] Determine the probability that component one in the system is still working at time given the system is working at time What is the limiting value as t_0? 2 marks] Suppose now that the system has one spare component which can be used to replace component in the failed system_ The time to failure ofthe spare cOm- ponent is also an exponential distribution with mean of one year._ Consider the following repair regime As soon as the system fails the one of the failed components in the system is replaced by the spare and the system continues to function as long as there are at least two components are working: Simu- late the processes in R or MATLAB You may wish to use the function sort in MATLAB or R Provide histogram of the failure times for the repaired system and estimate of the mean time to failure for the repaired system_ Supply VOuT code. marks