4.5. There are two common types of failure to a critical electronic element of s
ID: 3042988 • Letter: 4
Question
4.5. There are two common types of failure to a critical electronic element of some machinery: either component A or component B may fail. If either component fails, the machinery goes down. Component A fails according to a Poisson process with mean rate 1.1 failures per shift. (The company operates 24/7 using eight-hour shifts.) Component B fails according to a Poisson process with a mean rate of 1.2 failures per day. (a) What is the probability that there will be exactly five failures of the machine within a given day? (b) What is the probability that there will be no more than one failure of the machine during the next shift? e tis now noon and the most recent failure occurred four hours earlier. What the probability that the next machine failure will occur before 6PM?Explanation / Answer
Here for Component A expected number of failures per days = 1.1 * 3 = 3.3 failures per day
for Component B, expected number of failure per days = 1.2 failures per day
(a) So, as if component A or compoennt B fails, whole machinery will fail.
So failure rate for machinery will be 3.3 + 1.2 = 4.5 failure per day
So, if X is the number of machine failure in a random day.
Pr(X = 5) = POISSON (X = 5; 4.5) = e-4.5 4.55 /5! = 0.1708
(b) Here now question asks for one shift. as there are 3 shifts per day so expected number of failure per shift = 4.5/3 = 1.5 failures per shift.
So, if X is the number of machine failure in a random shift
Pr(X <= 1) = Or(0 or 1 failure) = POISON (X = 0 ; 1.5) + POISSON (X = 1; 1.5) = e-1.5 1.50/0! + e-1.5 1.51/1! = 0.2231 + 0.3347 = 0.5587
(c) Here most recent failures occurs four hours earlier but as the given distribution is poisson and its memoryless so it doesn't effect that what would happened in past.
so till 6 PM, there are 6 hours.
Expected number of failures in 6 hours = 4.5 = 1.125 per 6 hours
Pr(next machine failure before 6 PM) =1 - Pr(no machine failure before 6 PM) = 1 - e-1.125 * 1.1250/0! = 1 - 0.3247 = 0.6753
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