Final answer:
To calculate the Bayes estimate for λ, combine the exponential prior distribution of λ with the likelihood function informed by the observed average time between failures, using Bayes' Theorem. The Bayes estimate is the expected value of the posterior distribution of λ.
Step-by-step explanation:
The question asks us to find the Bayes estimate for λ, the parameter of an exponential distribution that represents the time between failures of a machine, given that the prior distribution for λ is exponential with a mean of 100 hours. Two machines are observed with an average time between failures (x) of 1125 hours.
Bayes' theorem combines the prior distribution of λ, the likelihood function based on the observed data, and the evidence to provide a posterior distribution for λ. In this case, we have an exponential prior for λ and the likelihood function also being exponential due to the nature of the time between failures.
The Bayes estimator for λ is found by calculating the expected value of the posterior distribution, which often involves mathematically combining the prior distribution's parameters with the observed data.