4. Let y1θ ~iid Uniform (0,0), for i-1, n, Assume the prior distribution for θ to , be Pareto(a, b), where p()b1 for 0> a and 0 otherwise. Find the posterior distribution of θ.
Let X1,…Xn ~ iid Gamma (α, θ) where the α is known and interested in the rate parameter θ, and we chosen a prior θ~ Gamma (3, 1). Find the posterior distribution
Let Xi iid∼ N(0, θ) for i = 1, ..., n.
a) Find the MLE for θ. Call it
b) Is biased?
c) Is
consistent?
d) Find the variance of
(e) What is the asymptotic distribution of ?
(10 points) Let ynumber of patients to seek medical care at a particular urgent care facility in a one day period, and assume that, conditionally on θ, y has a Poisson(θ) distribution, so that E(y|θ) var(y(9) t θ ~ Gamma(α, β), where α > 0 and β > 0 are both fixed and known. -0. Further suppose tha Find the prior predictive mean and variance for the number of patients today, E(y) and var(y)
2) Let Yi,., Yn be iid N(a,a2). Let a~ known Find the posterior distribution p(u|3y). This distribution will depend N (8, T2) and trcat o2, 6, and r2 as fixed and on 2, 6, and 2. Calculations are tedious here. Use the hints given in class and follow through
2) Let Yi,., Yn be iid N(a,a2). Let a~ known Find the posterior distribution p(u|3y). This distribution will depend N (8, T2) and trcat o2, 6, and r2 as fixed and...
Let Xi, , X. .., Exp(β) be IID. Let Y max(Xi, , h} Find the probability density function of Y. İlint: Y < y if and only if XS for i 1,,n.
iid 14 marksAssume that e Denote T 4i Gamma(k, A) and X1,... , X,,e Poisson(0) (a) [4 marks Show that the posterior distribution of 0 is Gamma(nTk, n ). (b) [4 marks Find the probability function of the marginal distribution of Y = nX. (Note that the conditional distribution of on Y is not the same X1, ..., Xn.) as on
iid 14 marksAssume that e Denote T 4i Gamma(k, A) and X1,... , X,,e Poisson(0) (a) [4 marks Show...
Problem 3. (30 pts) Let W, i = 1,...,n be iid Exp(6.), Vi, i = 1,...,m be iid Exp(02), and two samples are independent, fw(w) dhe , fv(u) = bene (c) Provide the MLE for (6,62) = (0 - 0). (d) If a UMVUE exists for (61 – 62), provide it; otherwise explain why it does not exist.
Let Xi....,Xn,..., ~iid Exp(1) and let Yn) be the sample maximum of the first n observations. Show that the limiting distribution of Zn-(Y(n)-log n) has CDF F(z) exp{-e-*), z є R.
One side concept introduced introduced in the second Bayesian lecture is the conjugate prior. Simply put, a prior distribution π (0) is called conjugate to the data model, given by the likelihoodfunction L (Xi θ if the posterior distribution π (ex 2, , . , X ) is part of the same distribution family as the prior. This problem will give you some more practice on computing posterior distributions, where we make use of the proportionality notation. It would be...