4. Let X follow the exponential distribution for given θ > 0 and assume that θ follows the discre...
Let Xi , i = 1, · · · , n be a random sample from Poisson(θ) with pdf f(x|θ) = e −θ θ x x! , x = 0, 1, 2, · · · . (a) Find the posterior distribution for θ when the prior is an exponential distribution with mean 1; (b) Find the Bayesian estimator under the square loss function. (c) Find a 95% credible interval for the parameter θ for the sample x1 = 2, x2...
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
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.
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?
Problem 2.2 Suppose observations will be taken from a uniform distribution on the interval [e-0.5, e+ 0.5] where the value of θ is unknown, and the prior distribution of θ is a uniform distribution on the interval [10,20] a) If a single observation is selected and yields x-11, determine the posterior distribution of. ξ(9111) b) Ifn - 4 bservations are selected and yield data: 11, 11.5, 11.7, 10.9 then determine the posterior distribution of 0. ξ(θ| 11,1 15,1 1.7, 10.9)...
2. Suppose that X|θ ~ U(0.0), the uniform distribution on the interval (09). Assuming squared error loss, derive that Bayes estimator of θ with respect to the prior distribution P(α.θο), the two-parameter Pareto model specified in (3.36), first by explicitly deriving the marginal probability mass function of X, obtaining an expression for the posterior density of θ and evaluating E(θ x) and secondly by identifying g(θ|x) by inspection and noting that it is a familiar distribution with a known mean.
Let ?1, … , ?10 are identically independently distributed (iid) with an Exponential model with parameter ( λ ), a) Compute the likelihood function (LF). b) Adopt the appropriate conjugate prior to the parameter λ (Hint: Choose hyperparameters optionally within the support of distribution). c) Using (a) and (b), find the posterior distribution of λ. d) Compute the minimum Bayesian risk estimator of λ.
1. A certain continuous distribution has cumulative distribution function (CDF) given by F(x) 0, r<0 where θ is an unknown parameter, θ > 0. Let X, be the sample mean and X(n)max(Xi, X2,,Xn). (i) Show that θ¡n-(1 + )Xn ls an unbiased estimator of θ. Find its mean square error and check whether θ¡r, is consistent for θ. (i) Show that nX(n) is a consistent estimator of o (ii) Assume n > 1 and find MSE's of 02n, and compare...
A (3 pt) Let Xi, ,X, are drawn from the distribution ftheta(z) = F 404 (r+0) , for 0 < x < oo and 0 < θ < oo. We define Y = 3X an estimator for θ. Verify whether this estimator is unbiased? Find the MSE of Y. Hint: E(x)E(X B (3 pt) Let X,.., X, are drawn from the distribution fo) for O < x < 00 and 0 < θ < oo. We define Y = 2X...
C3.)Let f(z,0) (1/θ)2(1-0)/0, 0 < x < 1,0 < θ < oo. . Find the maximurn likelihood estimator of θ. Show that the maximurn likelihood estimator is unbiased to θ