Let X1, x2 ..... Xn be the iid Bernoulli (p). Then Y = ? (i=1 to n) is Binomial (n, p). Based on these pieces of information: (a) Find the posterior distribution of p given Y. (b) Does the prior distribution of p a conjugate family? (c) Find the Bayes estimator of p, P^hat_B
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Wei Z.
Let X1,...,Xn be a random sample from the Geometric distribution with pmf P(X=x) = (1-p)^x p, x= 0,1,2,... The mean of this distribution is (1-p)/p. (a) Find the estimator for p using the method of moments. (b) Find the MLE of p. (c) Consider a Beta prior on p, i.e., p ~ Beta(α,β). Find the posterior distribution of p. (d) What is the Bayes estimator of p under squared error loss? Denote it by p̂B. (e) What happens to p̂B if both α and β go to 0?
Sri K.
3. (a) Suppose that X_1, ..., X_n form a random sample from a Bernoulli distribution with parameter p. Find the MLE of p. (b) Suppose that X follows a Bernoulli distribution with parameter p in [0, 1]. Find the MLE of p. (c) Suppose that X follows a Bernoulli distribution with parameter p in [0.3, 0.5]. Now given the data, I know X takes the value of 1. Find the MLE of p.
Jacob F.
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