Question

The life times, YY in years of a certain brand of low-grade lightbulbs follow an exponential distribution with a mean of 0.55 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.

Answer 1

We are given the lifetime distribution here as:

as mean is reciprocal for parameter of the exponential distribution.

The probability of success now is computed here as:

a) The probability that the first success occurs in the fifth observation is computed here as:
= Probability of no success in the first 4 observations * Probability that fifth observation is a success

= (1 - 0.1623)⁴*0.1623

= 0.0799

Therefore 0.080 is the required probability here.

b) Given that first success occurred in 3rd observation, probability that the second success occurs in the 8th observation is computed here as:
= Probability of no success from 4th to 7th observations * Probability of success in the 8th observation

= Probability that the first success occurs in the 5th observation

= 0.080

Therefore 0.080 is the required probability here.

c) Probability that the first success occurs in odd numbered observation is computed here as:
= P(X = 1) + P(X =3) + P(X = 5) + ......... Infinity

= p + (1-p)²p + (1-p)⁴p + .... + Infinity

Therefore 0.54 is the required probability here.