Problem 4 - Bayesian inference with uniform prior The data are 21:n, the model is Normal(μ, σ*), with σ2 known. The problem is to obtain the posterior distribution of μ, p(p xỉ n, σ*)p(μ|xì n, σ2) wh...
5. Let y|μ ~ N(μ, φ), where φ is known. There is no reliable prior information about the mean other than that it is expected to be a positive quantity. Therefore, use the improper prior distribution: p(p)-1 if (0,x) and 0 otherwise. Suppose we observe one y. Then, find the posterior mean of p. (obtain an explicit expression)
DISTRIBUTION OF SAMPLE VARIANCE:
Xn ~ N(μ, σ2), where both μ and σ are Problem 4 (25 points). Assume that Xi unknowin 1. Using the exact distribution of the sample variance (Topic 1), find the form of a (1-0) confidence interval for σ2 in terms of quantiles of a chi-square distribution. Note that this interval should not be symmetric about a point estimate of σ2. [10 points] 2. Use the above result to derive a rejection region for a level-o...
Consider the simplified Bayesian model for normal data The joint posterior pdf is ful, σ21 x)a(σ2,-/2-1 expl_jy.tx, _aPI The marginal posterior pdfs of μ and σ 2 can be obtained by integrating out the other variable (8.30) y@1 x) α (σ2)-m;,-1/2 expl-- Σ.-tri-x)2 (8.31) d. Let q1 and q2 be they/2 and 1-y/2 quantiles of (8.31). Show that the 1-γ credible interval (gi,q2) is identical to the classic confidence interval (5.19) (with ar replaced by y). Hence, a (1-α) stochastic...
Let X1,X2, , Xn be a random sample from a normal distribution with a known mean μ (xi-A)2 and variance σ unknown. Let ơ-- Show that a (1-α) 100% confidence interval for σ2 is (nơ2/X2/2,n, nơ2A-a/2,n).
Let X1,X2, , Xn be a random sample from a normal distribution with a known mean μ (xi-A)2 and variance σ unknown. Let ơ-- Show that a (1-α) 100% confidence interval for σ2 is (nơ2/X2/2,n, nơ2A-a/2,n).
Stat 5644 Spring 2019 Homework 4 Due 3/20 Problem :UMVUE via Rao-Blackwell, Lehmann-Scheffe, and Basu theorems This problem is on the estimation of a reliability function. Let Xi, , x, be IID from N(μ, σ2). Let Φ(-) be the c.d.f. of the standard normal distribution. Assume that σ2-of is known, for now. It is of interest to estimate the reliability number σο for some c, with )-1-(). c is called cut of point in reliability (a) Give, without any proof,...
Problem 4 True or False A Bookmark this page Instructions: Be very careful with the multiple choice questions below. Some are "choose all that apply," and many tests your knowledge of when particular statements apply As in the rest of this exam, only your last submission will count. 1 point possible (graded, results hidden) The likelihood ratio test is used to obtain a test with non-asymptotic level o True O False Submit You have used 0 of 3 attempts Save...