8. Let X,.. , Xn be a random sampl le from a uniform(O, 0) distributio n....
3. Suppose that Xi,.... Xn is a random sample from a uniform distribution over [0,0) That is, 0 elsewhere Also suppose that the prior distribution of θ is a Pareto distribution with density 0 elsewhere where θ0 > 0 and α > 1. (a) Determine (b) Show , θ > max(T1 , . . . ,Zn,%) and hence deduce the posterior density of θ given x, . . . ,Zn is (c) Compute the mean of the posterior distribution and...
Specifically, suppose that Xi, X2, .., Xn denote n payments, modeled as iid random variables with common Weibull pdf 0, otherwise, where m > 0 is known and θ is unknown. In turn, suppose that θ ~ IG(α, β), that is, θ has an inverted gamma (prior) pdf 0, otherwise (a) Prove that the inverted gamma IG(α, β) prior is a conjugate prior for the Weibull family above. (b) Suppose that m-2, α-05, and β-2. Here are n-10 insurance payments...
Problem 3.1 Suppose that XI, X2,... Xn is a random sample of size n is to be taken from a Bermoulli distribution for which the value of the parameter θ is unknown, and the prior distribution of θ is a Beta(α,β) distribution. Represent the mean of this prior distribution as μο=α/(α+p). The posterior distribution of θ is Beta =e+ ΣΧ, β.-β+n-ΣΧ.) a) Show that the mean of the posterior distribution is a weighted average of the form where yn and...
3. Let Xi, , Xn be a random sample from a Poisson distribution with p.m.f Assume the prior distribution of Of λ is is an exponential with mean 1, i.e. the prior pdi g(A) e-λ, λ > 0 Note that the exponential distribution is a special gamma distribution; and a general gamma distribution with parameters α > 0 and β > 0 has the pd.f. h(A; α, β)-16(. otherwise Also the mean of a gamma random variable with the pd.f.h(Χα,...
Let X1, . . . , Xn be a random sample following Gamma(2, β) for some unknown parameter β > 0. (i) 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). (j) What is the posterior Bayes estimator of β assuming squared error loss?
Let the random sample X1, . . . , Xn be taken from the Binomial distribution with parameter θ, which is unknown and must be estimated. Let the prior distribution of θ be the beta distribution with known parameters α > 0 and β > 0. Find the Bayes risk and the Bayes estimator using squared error loss. estimator of θ.
Please answer the following question and show every step. Thank you. Let Xi,..,Xn be a random sample from a population with pdf 0, x<0, where θ > 0 is unknown. (a) Show that the Gamma(a, b) prior with pdf 0, θ < 0. is a conjugate prior for θ (a > 0 and b > 0 are known constants). (b) Find the Bayes estimator of θ under square error loss. (c) Find the Bayes estimator of (2π-10)1/2 under square error...
Let Xi, c〉0. Xn be i.i.d. from the Pareto distribution Pa(θ.e), θ 〉 0 Derive a ÙNIP test of size α for testing Ho : θ-Bo, c co versus 0, C > Co Let Xi, c〉0. Xn be i.i.d. from the Pareto distribution Pa(θ.e), θ 〉 0 Derive a ÙNIP test of size α for testing Ho : θ-Bo, c co versus 0, C > Co
Let Xi, c〉0. Xn be i.i.d. from the Pareto distribution Pa(θ.e), θ 〉 0 Derive a ÙNIP test of size α for testing Ho : θ-Bo, c co versus 0, C > Co Let Xi, c〉0. Xn be i.i.d. from the Pareto distribution Pa(θ.e), θ 〉 0 Derive a ÙNIP test of size α for testing Ho : θ-Bo, c co versus 0, C > Co
1. Let X1, ..., Xn be a random sample from a distribution with the pdf le-x/0, x > 0, N = (0,00). (a) Find the maximum likelihood estimator of 0. (b) Find the method of moments estimator of 0. (c) Are the estimators in a) and b) unbiased? (d) What is the variance of the estimators in a) and b)? (e) Suppose the observed sample is 2.26, 0.31, 3.75, 6.92, 9.10, 7.57, 4.79, 1.41, 2.49, 0.59. Find the maximum likelihood...