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 0.75 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

P(X>1)=e-1/0.75 =0.2636

a)P(first success in fifth observation)=(1-0.2636)4*0.2636=0.078

b) P(X2=8|X1=3)=P(X=5)=(1-0.2636)4*0.2636=0.078

c)P(success on odd number)=0.2636+(1-0.2636)2*0.2636+(1-0.2636)4*0.2636+(1-0.2636)6*0.2636+...

=0.2636/(1-(1-0.2636)2)=0.58

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