Suppose that a regular light bulb, a long-life light bulb, and an extra-long-life light bulb are being tested. The lifetime X1 of the regular bulb has the exponential distribution with mean θ, the lifetime X2 of the long-life bulb has the exponential distribution with mean 2θ, and the lifetime X3 of the extra-long-life bulb has the exponential distribution with mean 3θ.
a. Determine the M.L.E. of θ based on the observations X1, X2, and X3.
b. Let ψ = 1/θ, and suppose that the prior distribution of ψ is the gamma distribution with parameters α and β. Determine the posterior distribution of ψ given X1, X2, and X3.