letx1xnbe a random sample following gamma2 for some unknownparameter 0 a what is the distributio

(a) The distribution of $sum_{i=1}^n X_i$ is Gamma$(2n, beta)$ since the sum of independent Ga

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Let X1,...,Xn be a random sample following Gamma(2, β) for some unknown parameter β > 0. (a) What is the distribution of ∑ni=1Xi? You don't have to prove it. (b) Apply WLLN on the sample mean X̄n = ∑ni=1Xi/n. State the result. (c) What does X̄2n-3 log X̄n converge to in probability? Explain. (d) Apply CLT on X̄n. State the result. (e) Assume β = 2, use normal approximation to calculate P(1 ≤ ∑ni=1Xi ≤ 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 β^2? Explain. (h) Here's the observed data: x1 = 3, x2 = 10, x3 = 5. Find an estimate for β using the method of moments. (i) Here's another observed data set: x1 = 3, x2 = 10, x3 = 5000. Find an estimate for β using the 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?

          Let X1,...,Xn be a random sample following Gamma(2, β) for some unknown parameter β > 0.

(a) What is the distribution of ∑ni=1Xi? You don't have to prove it.
(b) Apply WLLN on the sample mean X̄n = ∑ni=1Xi/n. State the result.
(c) What does X̄2n-3 log X̄n converge to in probability? Explain.
(d) Apply CLT on X̄n. State the result.
(e) Assume β = 2, use normal approximation to calculate P(1 ≤ ∑ni=1Xi ≤ 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 β^2? Explain.
(h) Here's the observed data: x1 = 3, x2 = 10, x3 = 5. Find an estimate for β using the method of moments.
(i) Here's another observed data set: x1 = 3, x2 = 10, x3 = 5000. Find an estimate for β using the 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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