LetX1,...,Xnbe a random sample following Gamma(2,β) for some unknownparameterβ >0. (a) What is the distribution of∑ni=1Xi?...

LetX1,...,Xnbe a random sample following Gamma(2,β) for some unknownparameterβ >0.

(a) What is the distribution of∑ni=1Xi? You don’t have to prove it.

(b) Apply WLLN on the sample mean ̄Xn=∑ni=1Xi/n. State the result.

(c) What does ̄X2n−3 log ̄Xnconverge to in probability? Explain.

(d) Apply CLT on ̄Xn. State the result.

(e) Assumeβ= 2, use normal approximation to calculate P(1≤∑ni=1Xi≤3).Express results in terms of the CDF ofN(0,1), i.e., Φ(·).

(f) What is the MLE ofβ?

(g) What is the MLE ofβ2? Explain.

(h) Here’s the observed data:x1= 3,x2= 10,x3= 5. Find an estimate forβusing method of moments.

(i) Here’s another observed data set:x1= 3,x2= 10,x3= 5000. Find an estimateforβusing method of moments.

(j) Now let’s think like a Bayesian. Consider a prior distribution ofβ∼Gamma(a,b)for somea,b >0. Derive the posterior distribution ofβgiven (X1,...,Xn) =(x1,...,xn).

(k) What is the posterior Bayes estimator ofβassuming squared error loss?

Expert Solution

orchestra answered 2 years ago

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