Let Y be the nth order statistic of a random sample of size n from Unif(0, 0). Let 0 has the prior pdf h(0) = BaB /0ß+1, 0 > a > 0, where a > 0, ß > 0. Find the Bayes estimator of 0 inder the squared orror loss

A random sample of n, 20 winter days in Denver gave a sample mean pollution index x, = 43. Previous studles show that o,- 10. For Englewood (a suburb of Denver), a random sample of n, - 19 winter days gave a sample mean pollution Index of x, = 37. Previous studies show that g, -13. Assume the pollution Index is normally distributed in both Englewood and Denver. Do these data indicate that the mean population pollution index of Englewood is different (either way) from that of Denver in the winter? Use a 1% level of significance. (a) what is the level of significance? State the null and alternate hypotheses. (b) what sampling distribution will you use? What assumptions are you making? O The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations. O The Student's t. We assume that both population distributions are approximately normal with known standard devtations. O The standard normal. We assume that both population…

Let Y₂ represent the ith normal population with unknown mean 4, and unknown variance of for i=1,2. Consider independent random samples, Y₁₁, Yi2,,Yin, of size ni, from the ith population with sample mean Y, and sample variance S?=²-1₁-1(Y - Y₁². (a) What is the distribution of Y;? State all the relevant parameters of the distribution. (b) Find a level a test (that is, the rejection region) for testing Ho: ₁ = o versus Ha Hiio when of is unknown and n; is small. (c) In the context of the test in part (b), state the Type I error and give a probability statement for the level of significance, a.

Let Y1, . . . , YN be a random sample from the Normal distribution Yi ∼ N(ln β, s2) where s^2is known.Find the maximum likelihood estimator of b from first principles.Find the Score function, the estimating equation and the information matrix using GLM

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Q4 Three electron emitters produce electron beams with changing kinetic energies that are uniformly distributed in the ranges [3,7+45], [2,5+45], and [4,10+45] (Roundup after multiplying by f). Let Y denote the total kinetic energy produced by these electron emitters. (a) Suppose that the three beam energies are independent. Determine the mean and variance of Y . (b) Suppose that the covariance between any two beam energies is -0.5. Determine the mean and variance of Y . (c) Compare and comment on the results in parts (a) and (b).

In N(5,4): Round Z-scores to 2 places after the decimal and use rounded value to find the data value in N(5,4). If P(X a) = .02, then the z-score for a = If P(b

Let Y, represent the ith normal population with unknown mean 4, and unknown variance of for i=1,2. Consider independent random samples, Ya, Ya, Yin, of size n,, from the ith population with sample mean Y, and sample variance S?=₁1(Y-₁². (h) Find the standard error of U₂ in part (g), assuming that of = 0² = 0². (i) Discuss how the distribution of Y₁ - Y₂ can be used to test the equality of the two population means, #₁1 and 12, when o=o=o² is known. (i) Define appropriate rejection regions, in terms of Y₁-Y₂, for testing Ho: #₁ = 1₂ against a two-sided alternative hypothesis at the a level of significance.

Q11. Let X1,X 2,.,X, be a random sample from a Poisson distribution with parameter 2. (a) Find the Cramer-Rao lower bound for the variance of any unbiased estimator of 1. (b) Find the maximum likelihood estimator (m.l.e.) of å. (c) Is this m.l.e. unbiased and does its variance attain the Cramer-Rao lower bound?

Show that the mean of a random sample of size n from an exponential population is a minimum variance unbiased estimator of the parameter 0.