Answer:
Using the notation of the lecture note, assume that the means satisfy the condition that μ-m + (b-1)d-μ2-d . . .-μ,-d....
1. (40) Suppose that X1, X2, Xn forms an independent and identically distributed sample from a normal distribution with mean μ and variance σ2, both unknown: 2nơ2 (a) Derive the sample variance, S2, for this random sample. (b) Derive the maximum likelihood estimator (MLE) of μ and σ2 denoted μ and σ2, respectively. (c) Find the MLE of μ3 (d) Derive the method of moment estimator of μ and σ2, denoted μΜΟΜΕ and σ2MOME, respectively (e) Show that μ and...
Let independent random samples, each of size n, be taken from the k normal distributions with means u cd [j - (k 1)/2], j = 1, 2,..., k, respectively, and common variance o2. Find the maximum likelihood estimators of c and d
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3. Solve the following by hand, and by using either R or SAS: Let μ1-H2,H3 be, respectively, the means of three normal distributions with a common but unknown variance σ2. In order to test, at the 0-: 0.05 significance level, the hypothesis Ha : μι μ2 against all possible alternative hypotheses, we take an independent random sample of size 4 from each of these distributions. Determine whether...
I1. Follow the steps below to show that the pooled estimator $p is an unbi- ased estimator for the common standard deviation of two independent sam ples Let Yi, Yi2, ..., Yini denote the random sample of size n from the first population with population mean μ| and population variance σ, and let Y21, Y22, ..., Y2na denote an independent random sample of size n2 from the second population with population mean μ2 and population mean ơ3. Sup- pose that...
5. Let 11,D, , , ,Zn and yı, y2, . . . , ym denote independent observed random samples of size n and m taken from two normally distributed populations with the same mean μ but different variances σ and σ . lihood estimator for the common mean μ based on the combined sample Find the maximum like . Is pmle unbiased? Find the variance of nle. - Define the following estimator n+ m Is μ unbiased. Find the variance...
6. Consider the following sample: Xi = -2, X2 = 12. X7-1.5, Xs -0.5, a. Estimate the population mean, μ, using an analogical estimator. b. Estimate the population variance. ơ2, using a biased and an unbiased estimator. c. Assuming that the random sample is drawn from a normal population with known variance, σ2-4, construct a 95% confidence interval for the population mean. d. Assuming that the random sample is drawn from a normal population with unknown variance, σ2, construct a...
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...
Question 10
RD 1 (X-μ)/μ|. Show that (5.28) 9. See Problem 5.8. Compute the signal-to-noise ratio r for the random variables from the fol. lowing distributions: (a) P(A), (b) E(n, p), (c) G(p), (d) Γ(α, β), (e) W (α, β), (f) LNue). and (g) P(α,0), where α > 2. 10. Let X and F be the sample means from two independent samples of size n from a popu- lation with finite mean μ and variance σ. Use the Central Limit...
Assume that class grades follow a normal distribution of mean μ = 75 and the variance σ2 =144. a) Find the probability that an individual's grade is greater than 81. b) What should be the interquartile range? c)Suppose you select at random (and independently) 10 students. What is the probability that only two of these students have a grade greater than 75? d) If you draw a sample of size n = 10 from the population of grades described in...
R1. Suppose X is a continuous RV with E(X-μ and Var(X-σ2 where both μ and σ are unknown. Note that X may not be a normal distribution. Show that X is an asymptotically unbiased estimator for μ2. (This problem does not require the computer.) R2. Let X ~ N(μ 10.82). Following up on R1, we will be approximating μ2, which we can see should be 100, For now, let the sample size be n 3. Pick 3 random numbers from...