The life times, Y in years of a certain brand of low-grade lightbulbs follow an

Question

The life times, Y in years of a certain brand of low-grade lightbulbs follow an exponential distribution with a mean of 0.65 years. A tester makes random observations of the life times of this particular brand of lightbulbs and records them one by one as either a success if the life time exceeds 1 year, or as a failure otherwise.

Part a) Find the probability to 3 decimal places that the first success occurs in the fifth observation.

Part b) Find the probability to 3 decimal places of the second success occurring in the 8th observation given that the first success occurred in the 3rd observation.

Part c) Find the probability to 2 decimal places that the first success occurs in an odd-numbered observation. That is, the first success occurs in the 1st or 3rd or 5th or 7th (and so on) observation.

Explanation / Answer

probability of success =P(X>1) =e-1/0.65 =0.2147

a) probability   that the first success occurs in the fifth observation =P(til 4th  observation no success and on 5th  observation success occurs)=(1-0.2147)4*0.2147 =0.082

b) probability of the second success occurring in the 8th observation given that the first success occurred in the 3rd observation

=P( next 4  observation from 4th to 7th no success and on 8th  observation success occurs)

==(1-0.2147)4*0.2147 =0.082

c)

probability of success on odd numbered observation =P( success on 1st observation)+P( on first 2 trail no success and success on 3rd observation)+P(on first 4 trail no success and success on 5th observation)+..........

=0.2147+(1-0.2147)2*0.2147+(1-0.2147)4*0.2147+(1-0.2147)6*0.2147+..............

=0.2147/(1-(1-0.2147)2) =0.56

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.