A nonreplacement test was carried out on 100 electronic components with a known

Question

A nonreplacement test was carried out on 100 electronic components with a known constant failure rate. The history of failures was as follows: 1^st failure after 93 h 2^nd failure after 1, 010 h 3^rd failure after 5,000 h 4^th failure after 28,000 h 5^th failure after 63,000 h The testing was discontinued after the fifth failure. If we can assume that the test gives an accurate estimate of the failure rate, determine the probability that one of the components would last for (a) 10^5 h (b) 10^6 h. In Excel, plot the results of the test with failure number on the y-axis and time on the x-axis. Notice the exponential nature of the failures. Use the last three failures to determine a constant failure rate. Solve the question using this failure rate. Include your Excel graph with your written calculations.

Explanation / Answer

Let X = time to failure (life) of the electronic component.

Then, X ~ Exp(), where = average life, and its pdf (probability density function) of X is given by f(x) = (1/)e-x/, 0 x < ………………….....................................………………………(1)

CDF (cumulative distribution function), F(t) = P(X t) = 1- e-t/ ………………………….…(2)

From (2), P(X > t) = e-t/ ………………………….………………………..……………………(3)

From the failure time of the last three given components, = (5000 + 28000 + 63000)/3 = 32000.

Part (a)

Probability that a component will last for 105 hours = P(X > 105)

= e-100000/32000 [vide (3) under Back-up Theory]

= e-3.124 = 0.0440 ANSWER

Part (b)

Probability that a component will last for 106 hours = P(X > 106)

= e-1000000/32000 [vide (3) under Back-up Theory]

= e-31.24 = 2.71x10-14ANSWER

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