Concept explainers
A system consists of two components connected in series. The system will fail when either of the two components fails. Let T be the time at which the system fails. Let X1, and X2 be the lifetimes of the two components. Assume that X1 and X2 are independent and that each has the Weibull distribution with α = 2 and β = 0.2.
a. Find P(X1 > 5).
b. Find P(X1 > 5 and X2 > 5).
c. Explain why the
d. Find P(T ≤ 5).
e. Let t be any positive number. Find P(T ≤ t), which is the cumulative distribution
f. Does T have a Weibull distribution? If so, what are its parameters?
a.
Find the value of
Answer to Problem 13E
The value of
Explanation of Solution
Given info:
The total number of components is 2. The random variable T is defined as the time at which the system fails. The random variables
Assume that the lifetimes are independent. The random variables
Calculation:
The random variables
Weibull distribution:
The probability density function of the Weibull distribution with parameters
The cumulative distribution function of the Weibull distribution with parameters
Substitute
Thus, the value of
b.
Find the value of
Answer to Problem 13E
The value of
Explanation of Solution
Calculation:
Here, the random variables
Then the joint probability density function is the product of the marginal, each of which is Weibull with parameters
From part (a),
Then,
Thus, the value of
c.
Explain the reason behind the event
Explanation of Solution
The random variable T is defined as the time at which the system fails.
The random variables
Thus, the lifetime of the system will be greater than 5 hours if and only if the lifetimes of both components greater than 5 hours.
Hence, the event
d.
Find the value of
Answer to Problem 13E
The value of
Explanation of Solution
Calculation:
The random variable T is defined as the time at which the system fails.
From part (b), the value of
From, part(c), it is clear that the event
Then,
Substitute
Thus, the value of
e.
Find the cumulative distribution function of T.
Answer to Problem 13E
The cumulative distribution function of T is,
Explanation of Solution
Calculation:
Here the random variables
Then the joint probability density function is the product of the marginal, each of which is Weibull with parameters
Substitute
Then,
From, part(c), it is clear that the event
Then,
Thus, the value of
f.
Check whether T has a Weibull distribution or not. Find the parameters if so.
Answer to Problem 13E
The random variable T follows Weibull distribution with parameter
Explanation of Solution
The cumulative distribution function of the Weibull distribution with parameters
From (e), the cumulative distribution function of T is,
Thus, the random variable T follows Weibull distribution with parameter
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