For normal data with unknown mean µ and σ^2, we use the prior r = 1/σ^2 ∼ Gam (α, β) and µ | r ∼ N(µ0,1/(rτ0)). Suppose that we believe that E(µ) = 4, Var(µ) = 10, E[r] = 3, Var(r) = 6.
Q1. What values should be selected for parameters µ0, τ0, α, β?
For normal data with unknown mean µ and σ^2, we use the prior r = 1/σ^2 ∼ Gam (α, β) and µ | r ∼ N(µ0,1/(rτ0)). Suppose...
with parameters α and β. 2. Yİ,%, , Y, are a random sample from the Gamma distribution (a) Suppose that α 4 is known and β is unknown. Find a complete sufficient statistic for β. Find the MVUE of β. (Hint: What is E(Y)?) (b) Suppose that β 4 is known and α is unknown. Find a complete sufficient statistic for a. with parameters α and β. 2. Yİ,%, , Y, are a random sample from the Gamma distribution (a)...
Suppose that the monthly return of stock A is approximately normally distributed with mean µ and standard deviation σ, where µ and σ are two unknown parameters. We want to learn more about the population mean µ, so we collect the monthly returns of stock A in nine randomly selected months. The returns are given (in percentage) as follows: 0.3, 1.3, 1.5, −0.6, −0.2, 0.8, 0.8, 0.9, −1.2 Answer the following questions about the confidence intervals for µ. (a) Construct...
Exercise 2. Let (an) be a sequence, and α, β ε R such that α β. Suppose there exists N N such that for all n2 N Then for allm2 N, Give an example demonstrating that it is not necessarily true that for all m2 N sup{an : n > m} < β Exercise 2. Let (an) be a sequence, and α, β ε R such that α β. Suppose there exists N N such that for all n2 N...
Suppose we are given two separate groups of data, y1 and y2, which are given by the following: y1: (y1,...yn)^T and y2: (yn+1,yn+2....yn+m)^T such that yi ~ N (µ, σ^2), i = 1,2,...,n and yi ~ N (µ*, σ^2*), i = n+1, n+2...,n+m. Assume all data are mutually independent and IID, and T = transpose. a) if σ^2 = σ^2* = (σ^2), where (σ^2) is known, find the posterior dist p(µ,µ*|y1,y2) and find bayes estimator for µ and µ* (assume...
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).
6. Suppose we observe Y,... Yn from a normal distribution with unknown parameters such that Y 24, s2 36, and n 15. (a) Find the rejection region of a level α-0.05 test of H0 : μ-20 vs. H1 : μ * 20. Would this test reject with the given data? (b) Find the rejection region of a level α -0.05 test of Ho : μ < 20 vs. H1 : μ > 20 would this test reject with the given...
4 and 5 samples, the other in small samples. Which is which? Explain. (d) Suppose we know that the 5 values are from a symmetric distribution. Then the sample median is also unbiased and consistent for the population mean. The sample mean has lower variance. Would you prefer to use the sample 4. Suppose Yi, Y, are iid r ables with E(n)-μ, Var(K)-σ2 < oo. For large n, find the approximate 5. Suppose we observe Yi...Yn from a normal distribution...
If n < 30 and σ İs unknown, then the 100(1-α)% confidence interval for a population mean is_ 7. assuming the population is normally distributed xtra 21 1(degrees of freedom = n-2) a. 1 (degrees of freedom-n-1) c. t(degrees of freedom-n-1) 1 d. x ± te |--| (degrees of freedom n-2) 1 (degrees of freedom = n-1) If n interval when the level of confidence is 99% 8. 16, then find the number ta2 needed in the construction of a...
2. Suppose Yi,.. narei normal random variables with normal distribution with unknown mean and variance, μ and or. Let Y-욤 Σ;..x. For this problem, you may not assume that n is large. (a) What is the distribution of Y? (b) what is the distribution of z-(yo), (en, (n-) (c) what is the distribution of (n-p? (d) What is the distribution of Justify your answer. (e) Let Zi-(ga)' + (-)' + (yo)", z2 = (속)' + (n-e)' what is the distribution...
Suppose that X1, X2, . . . , Xn is an iid sample of N (0, σ2 ) observations, where σ 2 > 0 is unknown. Consider testing H0 : σ 2 = σ 2 0 versus H1 : σ 2 6= σ 2 0 ; where σ 2 0 is known. (a) Derive a size α likelihood ratio test of H0 versus H1. Your rejection region should be written in terms of a sufficient statistic. (b) When the null...