3. (15 points) e time to failure of an electrical component is exponentially dis

Question

3. (15 points) e time to failure of an electrical component is exponentially distributed free of charge. a mean (u) of 4 years. Failed components under warranty are replaced Exponential distribution: f(t) e-At (pdf) Note: = 1 / F(t) = 1-e-At (cdf) (a) What is the probability the component will fail within the first 2 years? P( 2)=- (b) What is the probability the component will continue to operate for at least 6 years? P(26) = (c) If the manufacturer wishes to replace no more than 3% of the components free of charge, approximately what warranty period (in days) should be specified?

Explanation / Answer

(a) Here = 1/4 years-1

so we have to find that component will fail in 2 years.

P(y <=2) = 1 - e-(1/4) * 2 =1 - e-0.5 = 0.3935

(b) Now we have to find that

P(y >=4) = 1 - (1 - e-(1/4) * 6 ) = e-3/2 = 0.2231

(c) Here we have to find the time T above which there are only 3% chance that the product will fail.

P(y > T) = 0.03

e-(1/4) * T = 0.03

0.03 = e-T/4

ln (0.03) = -T/4

T = 14.026 years = 5120 days

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