A lifetime test is carried out on a particular type of componentA total of 15 components are observed, with failure times observed from t = 0 until t = 50.Ten components fail during this interval.The respective failure times are as follows:5.3, 5.1, 9.4, 6.9, 3.7, 5, 6, 8.8, 9.9, 7.7.The failure times are assumed to be iid Exponential with hazard rate λ.Determine the maximum likelihood estimate lambda.(Give answers to 3 decimal places)

Question 3. [9 Marks]. Recall that the exponential PDF has the form f (t; λ) = λe−λtwhere t, λ > 0 and λ is known as the rate parameter. Consider a job shop that consists of3 identical machines and 2 technicians. Suppose that, the amount of time each machineoperates before breaking down is exponentially distributed with rate parameter 0.1, and theamount of time a technician takes to fix a machine is exponentially distributed with rateparameter 0.4. Suppose that all the times to breakdown and times to repair are independentrandom variables and let X(t) be the number of machines which are operating at time t.(a) Determine the Q-matrix for this Markov process.(b) Write the forward equations involving the P ′0j (t), j = 0, 1, 2, 3, in terms ofpj (t) := P0j (t) = P(X(t) = j|X(0) = 0).(c) Obtain the equilibrium probabilities pj = limt→∞ pj (t).(d) What is the average number of busy technicians in the long-run? [Note: do not expectyour answer to be an integer.]

We observe 11 components, starting from time t = 0. The component life times are independent. Observation ends immediately following the 9th failure, which occurs at time t = 13.48. Give (correct to 2 decimal places) an appropriate estimate for F(13.48), using the Mean ranking metho

Consider a radioactive sample whose decay constant is λ. Initial (t=0) number of nuclei is N0. Find the total life of all the nuclei that decay in the interval of second and third half-lives (i.e.t=t1/2 to t=3t1/2)

The commulative distribution of a component is given as F(t) = 0.2e^(-0vt) of the range of the random variable T= 3 ≤ t ≤ αbr Determine the mean time to failure MTTF What is the probability that the component will fail 10hrs

3. At Kabwe General Hospital, an average of 6 accident victims arrive per night in the intensive care unit. However, arrangements have been made such that the intensive care unit is able to handle 8 accident victims per night. If this number is exceeded the patients will not get optimal treatment as the unit’s capacity will have been exceeded. For the purpose of this exercise, let’s assume that X = number of accident victims who arrive at the intensive care unit per night is Poisson distributed with λ = 6. a) Is the assumption that X = number of accident victims who arrive at the intensive care unit per night is Poisson distributed a valid one? Give augments in favour and against this assumption. b) Given a night, what is the probability that the unit’s capacity will be exceeded? c) Given 10 nights, what is the probability that the capacity of the unit will be exceeded? d) What is the probability of at most 8 accident victims arriving at the unit in a given night?