At time t = 0, 12 identical components are tested. The lifetime distribution of each is exponential with parameter λ. The experimenter then leaves the test facility unmonitored. On his return 24 hours later, the experimenter immediately terminates the test after noticing that y = 7 of the 12 components are still in operation (so 5 have failed). Derive the mle of λ. [Hint: Let Y = the number that survive 24 hours. Then Y~ Bin(n, p). What is the mle of p? Now notice that p = P(X; ≥ 24), where X, is exponentially distributed. This relates to p, so the former can be estimated once the latter has been.] (Round your answer to four decimal places.) λ =
At time t = 0, 12 identical components are tested. The lifetime distribution of each is exponential with parameter λ. The experimenter then leaves the test facility unmonitored. On his return 24 hours later, the experimenter immediately terminates the test after noticing that y = 7 of the 12 components are still in operation (so 5 have failed). Derive the mle of λ. [Hint: Let Y = the number that survive 24 hours. Then Y~ Bin(n, p). What is the mle of p? Now notice that p = P(X; ≥ 24), where X, is exponentially distributed. This relates to p, so the former can be estimated once the latter has been.] (Round your answer to four decimal places.) λ =
MATLAB: An Introduction with Applications
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ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![At time \( t = 0 \), 12 identical components are tested. The lifetime distribution of each is exponential with parameter \( \lambda \). The experimenter then leaves the test facility unmonitored. On his return 24 hours later, the experimenter immediately terminates the test after noticing that \( y = 7 \) of the 12 components are still in operation (so 5 have failed). Derive the mle of \( \lambda \). [Hint: Let \( Y \) = the number that survive 24 hours. Then \( Y \sim \text{Bin}(n, p) \). What is the mle of \( p \)? Now notice that \( p = P(X_i \ge 24) \), where \( X_i \) is exponentially distributed. This relates \( \lambda \) to \( p \), so the former can be estimated once the latter has been.]
(Round your answer to four decimal places.)
\(\hat{\lambda} =\) [textbox]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1665bdf4-5411-4ca1-a903-736ef5c9444c%2F8479f259-9242-4239-a841-ac1d3e983a11%2F1ijsltd_processed.png&w=3840&q=75)
Transcribed Image Text:At time \( t = 0 \), 12 identical components are tested. The lifetime distribution of each is exponential with parameter \( \lambda \). The experimenter then leaves the test facility unmonitored. On his return 24 hours later, the experimenter immediately terminates the test after noticing that \( y = 7 \) of the 12 components are still in operation (so 5 have failed). Derive the mle of \( \lambda \). [Hint: Let \( Y \) = the number that survive 24 hours. Then \( Y \sim \text{Bin}(n, p) \). What is the mle of \( p \)? Now notice that \( p = P(X_i \ge 24) \), where \( X_i \) is exponentially distributed. This relates \( \lambda \) to \( p \), so the former can be estimated once the latter has been.]
(Round your answer to four decimal places.)
\(\hat{\lambda} =\) [textbox]
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