Bulbs produced by a factory Fi have lifetimes (in months) distribution as Exp(1/3^i) for i=1, 2,3. A firm randomly procures a bulb from the firm and is found to be working after 27months. The probability that it was produced by the factory F3?

Question

Answer

Understanding the Problem
We have bulbs produced by three different factories, with their lifetimes following an exponential distribution:
- Factory F1: Lifetime ~ Exp(1/3)
- Factory F2: Lifetime ~ Exp(1/9)
- Factory F3: Lifetime ~ Exp(1/27)
A bulb is found to be functioning after 27 months. We need to determine the probability that this bulb was produced by factory F3.
Using Bayes' Theorem
To find this probability, we can apply Bayes' theorem:
- P(F3 | Working at 27 months) = P(Working at 27 months | F3) * P(F3) / P(Working at 27 months)
Calculating Probabilities
1. Prior Probabilities (Assuming equal likelihood of selecting any factory):
- P(F1) = P(F2) = P(F3) = 1/3
2. Likelihoods:
- P(Working at 27 months | F1) = e^(-27/3) = e^(-9)
- P(Working at 27 months | F2) = e^(-27/9) = e^(-3)
- P(Working at 27 months | F3) = e^(-27/27) = e^(-1)
3. Total Probability:
- P(Working at 27 months) = P(Working at 27 months | F1) * P(F1) + P(Working at 27 months | F2) * P(F2) + P(Working at 27 months | F3) * P(F3)
- = (e^(-9) * 1/3) + (e^(-3) * 1/3) + (e^(-1) * 1/3)
Final Calculation
Now, substituting back into Bayes' theorem:
- P(F3 | Working at 27 months) = (e^(-1) * 1/3) / [(e^(-9) * 1/3) + (e^(-3) * 1/3) + (e^(-1) * 1/3)]
This can be simplified to find the required probability.
Conclusion
Calculating the above expression provides the probability that the working bulb after 27 months was produced by factory F3.

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