Question

Let X1, X2, ..., Xn be the i.i.d. Bernoulli(p). Then Y = ? (i=1 to n) Xi is Binomial(n,p). We assume the prior distribution of p is Beta(a,b). Based on these pieces of information, answer the followings i. Find the maximum likelihood estimate (MLE) of p. Show it is equal to X?. ii. Derive the joint distribution of Y and p iii. Find the marginal pdf of Y iv. Find the posterior distribution of p given Y v. Does the prior distribution of p a conjugate family? vi. Find the Bayes estimator of p, p?B vii. Find the posterior variance of p. Show that using the prior variance, in the limit, the parameters of the prior distribution have no influence on the posterior distribution. viii. Using (v) show that the Bayes estimator of p is a weighted average of prior mean and sample mean, where weights are determined by a, b, and n. ix. Compare the maximum likelihood estimator of p and bayes estimator p. (Hint: use Mean squared error (MSE) as a measure of performance). x. Based on the MSEs of X? and p?B Figure 7.3.1. Comparison of MSE of p? and p?B for sample sizes n = 4 and n = 400 in Example 7.3.5 Which estimator is performing better than other and in what situations? (X) Find the 95% confidence interval of MLE of p. (Xi) Find the 95% credible interval of Bayes estimator of p. Derive an explicit expression of this interval, under Normality distribution of the posterior distribution of bayes estimator p. (Xii) compare both the intervals derived in (X) and (Xi).

          Let X1, X2, ..., Xn be the i.i.d. Bernoulli(p). Then Y = ? (i=1 to n) Xi is Binomial(n,p). We assume the prior distribution of p is Beta(a,b). Based on these pieces of information, answer the followings
i. Find the maximum likelihood estimate (MLE) of p. Show it is equal to X?.
ii. Derive the joint distribution of Y and p
iii. Find the marginal pdf of Y
iv. Find the posterior distribution of p given Y
v. Does the prior distribution of p a conjugate family?
vi. Find the Bayes estimator of p, p?B
vii. Find the posterior variance of p. Show that using the prior variance, in the limit, the parameters of the prior distribution have no influence on the posterior distribution.
viii. Using (v) show that the Bayes estimator of p is a weighted average of prior mean and sample mean, where weights are determined by a, b, and n.
ix. Compare the maximum likelihood estimator of p and bayes estimator p. (Hint: use Mean squared error (MSE) as a measure of performance).
x. Based on the MSEs of X? and p?B
Figure 7.3.1. Comparison of MSE of p? and p?B for sample sizes n = 4 and n = 400 in Example 7.3.5
Which estimator is performing better than other and in what situations?
(X) Find the 95% confidence interval of MLE of p.
(Xi) Find the 95% credible interval of Bayes estimator of p. Derive an explicit expression of this interval, under Normality distribution of the posterior distribution of bayes estimator p.
(Xii) compare both the intervals derived in (X) and (Xi).

Let X1, X2, ..., Xn be the i.i.d. Bernoulli(p). Then Y = ? (i=1 to n) Xi is Binomial(n,p). We assume the prior distribution of p is Beta(a,b). Based on these pieces of information, answer the followings
i. Find the maximum likelihood estimate (MLE) of p. Show it is equal to X?.
ii. Derive the joint distribution of Y and p
iii. Find the marginal pdf of Y
iv. Find the posterior distribution of p given Y
v. Does the prior distribution of p a conjugate family?
vi. Find the Bayes estimator of p, p?B
vii. Find the posterior variance of p. Show that using the prior variance, in the limit, the parameters of the prior distribution have no influence on the posterior distribution.
viii. Using (v) show that the Bayes estimator of p is a weighted average of prior mean and sample mean, where weights are determined by a, b, and n.
ix. Compare the maximum likelihood estimator of p and bayes estimator p. (Hint: use Mean squared error (MSE) as a measure of performance).
x. Based on the MSEs of X? and p?B
Figure 7.3.1. Comparison of MSE of p? and p?B for sample sizes n = 4 and n = 400 in Example 7.3.5
Which estimator is performing better than other and in what situations?
(X) Find the 95% confidence interval of MLE of p.
(Xi) Find the 95% credible interval of Bayes estimator of p. Derive an explicit expression of this interval, under Normality distribution of the posterior distribution of bayes estimator p.
(Xii) compare both the intervals derived in (X) and (Xi).

Added by Minhaz U.

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