Show that the gamma distribution is a conjugate prior for the exponential distribution. Suppose that the waiting time in a queue is modeled as an exponential random variable with unknown parameter λ, and that the average time to serve a random sample of 20 customers is 5.1 minutes. A gamma distribution is used as a prior. Consider two cases: (1) the mean of the gamma is 0.5 and the standard deviation is 1, and (2) the mean is 10 and the standard deviation is 20. Plot the two posterior distributions and compare them. Find the two posterior means and compare them. Explain the differences.
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