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Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi
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A system consists of two components connected in series.

Chapter 4, Problem 13E

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QUESTION:

A system consists of two components connected in series. The system will fail when either of the two components fails. Let  be the time at which the system fails. Let  and  be the lifetimes of the two components. Assume that \(X_{1}\) and  are independent and that each has the Weibull distribution with \(\(\alpha\)\) = 2 and \(\beta\) = 0.2
a. Find .
b. Find  and .
c. Explain why the event  is the same as the event (\(\(X_{1}\)\) >5 and \(X_{2}\) >5)
d. Find .
e. Let  be any positive number. Find , which is the cumulative distribution function of .
f. Does  have a Weibull distribution? If so, what are its parameters?

Equation transcription:

Text transcription:

X_{1}

X_{2}

\alpha

\beta

Questions & Answers

QUESTION:

A system consists of two components connected in series. The system will fail when either of the two components fails. Let  be the time at which the system fails. Let  and  be the lifetimes of the two components. Assume that \(X_{1}\) and  are independent and that each has the Weibull distribution with \(\(\alpha\)\) = 2 and \(\beta\) = 0.2
a. Find .
b. Find  and .
c. Explain why the event  is the same as the event (\(\(X_{1}\)\) >5 and \(X_{2}\) >5)
d. Find .
e. Let  be any positive number. Find , which is the cumulative distribution function of .
f. Does  have a Weibull distribution? If so, what are its parameters?

Equation transcription:

Text transcription:

X_{1}

X_{2}

\alpha

\beta

ANSWER:

Solution:

Step 1 of 5:

A system consist of two components which is connected in series.the system will fail if one of the two component fails. Let T be the lifetime of the system and X1 and X2 be the lifetime of each components. Here X1 and X2 are independent which follows weibull distribution with parameter and .

We have to find

  1. P(X1>5)
  2. P(X1>5 and X2>5)
  3. The event T>5 is as same as the event {X1>5 and X2>5}.
  4.  P(
  5. P, which is the cumulative density function of T.
  6. The distribution of T with parameters.

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