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A system consists of five components is connected in series as shown below. -1 42 43...
A system consists of five identical components connected in seriesas shown:As soon as one...
A system consists of five identical components connected in series
as shown:As soon as one components fails, the entire system will fail.
Suppose each component has a lifetime that is exponentially
distributed with ? = 0.01 and that components fail independently of one another.
Define eventsAi= {ith
component lasts at least t hours}, i = 1, . . . , 5, so that the Ais
are independent events. Let X = the time at which the system failsthat is, the...
Problem #7: Suppose that 26% of all steel shafts produced by a certain process are nonconforming...
Problem #7: Suppose that 26% of all steel shafts produced by a certain process are nonconforming but can be reworked (rather than having to be scrapped). (a) In a random sample of 175 shafts, find the approximate probability that between 37 and 53 (inclusive) are nonconforming and can be reworked. (b) In a random sample of 175 shafts, find the approximate probability that at least 49 are nonconforming and can be reworked. Problem #8: A system consists of five components...
(12 points) Consider the system comprised of three components as shown below. Suppose The lifetime of...
(12 points) Consider the system comprised of three components as shown below. Suppose The lifetime of Component 1 is exponentially-distributed with parameter 11 = 1/10. • The lifetime of Component 2 is exponentially-distributed with parameter 12 = 1/20. • The lifetime of Component 3 is exponentially-distributed with parameter 13 = 1/15. The system is working if both (A) Component 1 is working, and (B) Component 2 or/and Component 3 is working. Compute the probability that the system is still working...
Question 4 [20 marks] A system consists of five components in two branches as shown in...
Question 4 [20 marks] A system consists of five components in two branches as shown in the following diagram: C-D-E- In other words, the system works if components A and B work or components C, D, and E work. Assume that the components fail independently with the following probabilities: P(A fails) = P(B fails) = 0.1 and P(C fails) = P(D fails) = P(E fails) = 0.2. (a) What is the probability that the system works? (b) Given that the...
5. Lec 17 function of pairs of R.V., 8 pts) Let X be the lifetime of a critical and expensive com...
5. Lec 17 function of pairs of R.V., 8 pts) Let X be the lifetime of a critical and expensive component in a system, which is exponentially distributed with mean 2 years. The system also has a cheaper backup component that can take over when the expensive component fails so that the system can provide continuous service while the more expensive system is being repaired. Let Y be the lifetime of the backup system, which is also exponentially distributed but...
Suppose a system of ive components Ai,1 Si S 5 is arranged as follows 2 Assum e the lifetime of each component is exponentially distributed with parameter) and the components function independently....
Suppose a system of ive components Ai,1 Si S 5 is arranged as follows 2 Assum e the lifetime of each component is exponentially distributed with parameter) and the components function independently. Let of the i-th component, that is the random variable defined by (Xi - t) means that the the i-th component stops working at time t. Saying that Xi has an exponenti distribution with parameter X means X, be the lifetime random variable and P(Xi s t)-1-e*. be...
4. Reliability of Systems - Take n components to have failure times Ti, T2, ..., Tn...
4. Reliability of Systems - Take n components to have failure times Ti, T2, ..., Tn If we construct a complex system out of these distribution of the failure time T of the entire svstem in terms of the distributions of Ti, T2, ..., Tn. There are two basic networks. In a series hookup, the system fails as soon as any one of the components fails. Hence T - min(T1, T2, ...,Tn). In a parallel hookup the system is operational...
A system consists of three components A, B and C, which fails independently with probabilities 0.2,...
A system consists of three components A, B and C, which fails
independently with probabilities 0.2, 03 and 0.2. Let X be the
total number of failed components.
(a) Find the probability distribution of X.
(b) What is the probability that at least one component is
working.
(c) Find E(X^3 − 1).
3. (7 points) A system consists of three components A, B and C, which fails independently with probabilities 0.2, 03 and 0.2. Let X be the total number...
Suppose 4 components of system all act independently, and each fails within a year 10% of...
Suppose 4 components of system all act independently, and each fails within a year 10% of the time. Find the probability none fails within a year Find the probability at least one fails within a year Find the probability all fail within a year. Find the probability at least one works the entire year.
upposea components of system all act independently, and each fails within a year 10% of the...
upposea components of system all act independently, and each fails within a year 10% of the time Find the probability none fails within a year Find the probability at least one fails within a year Find the probability all fail within a year. Find the probability at least one works the entire year.
A system consists of five identical components connected in series
as shown:As soon as one components fails, the entire system will fail.
Suppose each component has a lifetime that is exponentially
distributed with ? = 0.01 and that components fail independently of one another.
Define eventsAi= {ith
component lasts at least t hours}, i = 1, . . . , 5, so that the Ais
are independent events. Let X = the time at which the system failsthat is, the...
Problem #7: Suppose that 26% of all steel shafts produced by a certain process are nonconforming but can be reworked (rather than having to be scrapped). (a) In a random sample of 175 shafts, find the approximate probability that between 37 and 53 (inclusive) are nonconforming and can be reworked. (b) In a random sample of 175 shafts, find the approximate probability that at least 49 are nonconforming and can be reworked. Problem #8: A system consists of five components...
(12 points) Consider the system comprised of three components as shown below. Suppose The lifetime of Component 1 is exponentially-distributed with parameter 11 = 1/10. • The lifetime of Component 2 is exponentially-distributed with parameter 12 = 1/20. • The lifetime of Component 3 is exponentially-distributed with parameter 13 = 1/15. The system is working if both (A) Component 1 is working, and (B) Component 2 or/and Component 3 is working. Compute the probability that the system is still working...
Question 4 [20 marks] A system consists of five components in two branches as shown in the following diagram: C-D-E- In other words, the system works if components A and B work or components C, D, and E work. Assume that the components fail independently with the following probabilities: P(A fails) = P(B fails) = 0.1 and P(C fails) = P(D fails) = P(E fails) = 0.2. (a) What is the probability that the system works? (b) Given that the...
5. Lec 17 function of pairs of R.V., 8 pts) Let X be the lifetime of a critical and expensive component in a system, which is exponentially distributed with mean 2 years. The system also has a cheaper backup component that can take over when the expensive component fails so that the system can provide continuous service while the more expensive system is being repaired. Let Y be the lifetime of the backup system, which is also exponentially distributed but...
Suppose a system of ive components Ai,1 Si S 5 is arranged as follows 2 Assum e the lifetime of each component is exponentially distributed with parameter) and the components function independently. Let of the i-th component, that is the random variable defined by (Xi - t) means that the the i-th component stops working at time t. Saying that Xi has an exponenti distribution with parameter X means X, be the lifetime random variable and P(Xi s t)-1-e*. be...
4. Reliability of Systems - Take n components to have failure times Ti, T2, ..., Tn If we construct a complex system out of these distribution of the failure time T of the entire svstem in terms of the distributions of Ti, T2, ..., Tn. There are two basic networks. In a series hookup, the system fails as soon as any one of the components fails. Hence T - min(T1, T2, ...,Tn). In a parallel hookup the system is operational...
A system consists of three components A, B and C, which fails
independently with probabilities 0.2, 03 and 0.2. Let X be the
total number of failed components.
(a) Find the probability distribution of X.
(b) What is the probability that at least one component is
working.
(c) Find E(X^3 − 1).
3. (7 points) A system consists of three components A, B and C, which fails independently with probabilities 0.2, 03 and 0.2. Let X be the total number...
Suppose 4 components of system all act independently, and each fails within a year 10% of the time. Find the probability none fails within a year Find the probability at least one fails within a year Find the probability all fail within a year. Find the probability at least one works the entire year.
upposea components of system all act independently, and each fails within a year 10% of the time Find the probability none fails within a year Find the probability at least one fails within a year Find the probability all fail within a year. Find the probability at least one works the entire year.
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