Bayesian analysis of the exponential distribution A lifetime

Bayesian analysis of the exponential distribution A lifetime X of a machine is modeled by an exponential distribution with unknown parameter theta. The likelihood is p(x\\theta)=thetae-thetax for x LE 0, theta > 0 Show that the MLE is theta =1/x ,where x =1/n xi Suppose we observe X1= 5,x2=6,x3=4 (the lifetimes (in years) of 3 different iid machines). What is the MLE given this data? Assume that an expert believes theta should have a prior distribution that is also exponential p(theta) = Expon(theta\\lambda) Choose the prior parameter, call its lambda, such that E[theta]=1/3. Ga (theta| a,b) alpha thetaa-1e-thetab and its mean is a/b. What is the posterior, p(theta\\D, lambda )? Is the exponential prior conjugate to the exponential likelihood? What is the posterior mean, E [theta\\D, lambda ]? Explain why the MLE and posterior mean differ. Which is more reasonable in this example?

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