3. Show that if X θ ~ Norina!(μ,0) (Note: μ is known! θ is the unknown variance) and θ has a pdf ...
Two questions exist : ) if it has pdf. A railon variable X has the l'areio distril illi ribution with parameters m, a (m, α > 0 w 0 otherwise Show that if X has this Pareto distribution, then the random variable log(X/m) has the expo- nential distribution with parameter α Let X ~ Gamma(α, β), where α > 1 . Find E[1/X]. ) if it has pdf. A railon variable X has the l'areio distril illi ribution with parameters...
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...
2. 12 points Bjorn, a member of ABBA, knows that X (which is the amount an ABBA song brightens or worsens a person's day) has a Normal distribution with a mean of 0... and he's OK with that. What he wants to learn more about is the precision of X, which is the inverse of the variance. Bjorn thinks that the precision has a Gamma distribution as described below. Help Bjorn out by finding the posterior distribution for the percision....
5, (2 pt) Assume that the variance σ2 is known. Let the likelihood of μ oe i-1 Let θ' and θ', be distinct fixed values of θ so that Ω-10; θ-θ'), and let k be a positive number. Let C be a subset of the sample space such that () for each point z E C. (b) for each point C. L(0"a) Show that C is the best critical region of size α for testing: H0 : θ- 5, (2...
6.29. Let Xi, X2, , X,be a random sample from a gamma distribution with known parameter α-3 and unknown β > 0,' Discuss the construction of a confidence interval for B. Hint: what is the distribution of 2 Σ x/P Follow the procedure outlined in Exercise 6.28. 6.29. Let Xi, X2, , X,be a random sample from a gamma distribution with known parameter α-3 and unknown β > 0,' Discuss the construction of a confidence interval for B. Hint: what...
Let X,X,, X, be a random sample of size 3 from a uniform distribution having pdf /(x:0) = θ,0 < x < 0,0 < θ, and let):く,), be the corresponding order statistics. a. Show that 2Y, is an unbiased estimator of 0 and find its variance. b. Y is a sufficient statistic for 8. Determine the mean and variance of Y c. Determine the joint pdf of Y, and Y,, and use it to find the conditional expectation Find the...
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...
Recall that if X has a beta(a, B) distribution, then the probability density function (pdf) of X is where α > 0 and β > 0. In this problem, we are going to consider the beta subfamily where α-β θ. Let X1, X2, , Xn denote an iid sample from a beta(8,9) distribution. (b) The two-dimensional statistic nm 27 is also a sufficient statistic for θ. What must be true about the conditional distribution (c) Show that T* (X) is...
3x2 for 0 < x < θ and zero otherwise. With the parameter θ > 0. We wish to Consider the pdf,f(x) estimate θ using the sample maximum from a random sample (iid) of size n. 0n-maxi Xi. (hint: first find the CDF and PDF of the estimator) Show this estimator is consistent a. b. Show this estimator is biased C. Suggest a better estimator and show that it is UC d. Show that n(9-an) converges (using the original estimator,...
6. L , Xn be a random sample from a population with pdf et X1,. . . 9x1, xe (0,1), 0, otherwise, where θ E Θ (0.00) (a) Find a confidence interval for θ with confidence coefficient 1-α by pivoting a random variable based on statistic T(X,)--Σ-1 log Xi. (Use quantiles of chi-square distributions to express the confidence interval and use equal-tail confidence interval) (b) Find the shortest I-α confidence interval for θ of the form a/T, b/T, where T(X,)...