Reference no: EM132850992
Let X1, ..., Xn be a random sample following Gamma(2, β) for some unknown parameter β >0.
(a) What is the distribution of ∑n i=1 Xi? You don't have to prove it.
(b) Apply WLLN on the sample mean ¯Xn=∑n i=1 Xi/n. State the result.
(c) What does ¯X2 n - 3log ¯Xn converge to in probability? Explain.
(d) Apply CLT on ¯Xn. State the result.
(e) Assume β = 2, use normal approximation to calculate P(1≤∑n i=1 Xi≤3). Express results in terms of the CDF of N(0,1), i.e., Φ(·).
(f) What is the MLE of β?
(g) What is the MLE of β2? Explain.
(h) Here's the observed data: x1= 3, x2= 10, x3= 5. Find an estimate for β using method of moments.
(i) Here's another observed data set: x1= 3, x2= 10, x3= 5000. Find an estimate for β using method of moments.
(j) Now let's think like a Bayesian. Consider a prior distribution of β∼Gamma (a, b) for some a, b >0. Derive the posterior distribution of β given (X1, ..., Xn) = (x1, ..., xn).
(k) What is the posterior Bayes estimator of β assuming squared error loss?
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