3. Prove the theorem for t he normal conjugate distributi on Theorem. Suppose that Xi,... ,Xn...
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...
Suppose that Xi, X2, , xn is an iid sample from a U(0,0) distribution, where θ 0. În turn, the parameter 0 is best regarded as a random variable with a Pareto(a, b) distribution, that is, bab 0, otherwise, where a 〉 0 and b 〉 0 are known. (a) Turn the "Bayesian crank" to find the posterior distribution of θ. I would probably start by working with a sufficient statistic (b) Find the posterior mean and use this as...
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...
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).
8. Suppose that Xi,..., Xn is a sample from a normal population having unknown pa- rameters μ and σ2 a) Devise a significance level α test of the null hypothesis 0 versus the alternative hypothesis for a given positive value σ1. b) Explain how the test would be modified if the population mean μ were known in advance 8. Suppose that Xi,..., Xn is a sample from a normal population having unknown pa- rameters μ and σ2 a) Devise a...
2. Suppose that Xi, , Xn, n-: 25, form a random sample from a normal distribution with mean θ and variance 4. Consider the following hypotheses at α-0.05 Ho : θ-0 versus H1 : θ > 0. Derive the power function, π( 5), and evaluate it at θ--04,-02, 0,02, 0.4, 0.6, 0.8, 1. 2. Suppose that Xi, , Xn, n-: 25, form a random sample from a normal distribution with mean θ and variance 4. Consider the following hypotheses at...
Let Xi,, Xn be a random sample of size n from the normal distribution with mean parameter 0 and variance σ2-3. (a) Justify thatX X, has a normal distribution with mean parameter 0 and variance 3 /n, this is, X~N(0,3/m) (you can do it formally using m.g.f. or use results from normal distribution to justify (b) Find the 0.975 quantile of a standard normal distribution (you can use a table, software or internet to find the quantile). (c) Find the...
xercise 7.5: Suppose Xi, X2, ..., Xn are a random sample from the u distribution U(9-2 ,0+ ), where θ e (-00, Exercise 7.5: Suppose X1, X2, . .. , sufficient for θ. a) Show that the smallest and largest of Xi, ..., Xn are jointliy (b) If p@-constant, θ e (-00, oo), is the prior distribution of θ, find its posterior distribution xercise 7.5: Suppose Xi, X2, ..., Xn are a random sample from the u distribution U(9-2 ,0+...
8. Suppose that Xi,..., Xn is a sample from a normal population having unknown pa- rameters μ and σ2 a) Devise a significance level α test of the null hypothesis 0 versus the alternative hypothesis for a given positive value σ1. b) Explain how the test would be modified if the population mean μ were known in advance