**Problem 9.6**Suppose that \( Y_1, Y_2, \ldots, Y_n \) denote a random sample of size \( n \) from a Poisson distribution with mean \( \lambda \). Consider \( \hat{\lambda}_1 = (Y_1 + Y_2)/2 \) and \( \hat{\lambda}_2 = \overline{Y} \). Derive the efficiency of \( \hat{\lambda}_1 \) relative to \( \hat{\lambda}_2 \).

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Answered: 9.6 Suppose that Y₁, Y₂,... Y, denote a…

MATLAB: An Introduction with Applications

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ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

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9.3 Let Y₁, Y₂,..., Y, denote a random sample from the uniform distribution on the interval (0, 0+1). Let 0 =Y-; and O=Y). a Show that both ₁ and ₂ are unbiased estimators of 0. b Find the efficiency of 8₁ relative to ₂. n n+1
22. Given a random sample {(x₁, y₁), (x₂, y2), (xn. Yn)} such that Sxx = 107 and Syy = 54.75. If the regres line that relates the variables X and Y has a slope of 0.257, find the linear correlation coefficient r. A 0.069 B 0.359 C 0.083 D 0.453
Q 4.4. An individual picked at random from a population has a propensity to have accidents that is modelled by a random variable Y having the gamma distribution with shape parameter a and rate parameter ß. Given Y = y, the number of accidents that the individual suffers in years 1, 2, ..., n are independent random variables X₁, X2,... Xn each having the Poisson distribution with parameter y. (a) Write down a function f so that the joint distribution of Y, X₁, Xn can be described via P(a ≤ Y ≤ b, X₁ = k1, X₂ = k2 … . . Xn = kn) = [° ƒ (y, k1, k2, ... kn)dy and derive from this expression that, for your choice of f, Y has the Gamma distribution, and that conditionally on Y = y, X₁, X2,... Xn are independent, each having the Poisson distribution with parameter y.
2
1. Suppose a X value is produced by a random process that has a mean of 1 and a variance of 9, while a Y value is produced by a random process that has a mean of 9 and a variance of 1. Let Z = 3X - 2Y. Compute: a. the mean and variance of Z assuming X and X are independent. b. the mean and variance of Z assuming X and X are not independent and their correlation coefficient is 0.50.
4, 12. How Call v. 13. Suppose that two raters (Rater A and Rater B) each assign pirysica attractiveness scO = not at all attractive to 10 = extremely attractive) to a set of seven facial photogranhe 21 %3D Pearson's r can be used as an index of interrater reliability or agreement on quantitative ratings. A correlation of +1 would indicate perfect rank-order agreement between rate while an r of 0 would indicate no agreement about judgments of relative attractiveness An r of .8 to .9 is considered desirable when reliability is assessed. For this example, Rater A's score is the X variable and Rater B's score is the Y variable. The ratings are as followe F follows: Photo Rater A Rater B 1 3. 5. 8. 6. 7. 8. 6. 4. 6. 10 6. 7. 5. 4. a. Compute the Pearson correlation between the Rater A and Rater B attractiveness ratings. What is the obtained r value? b. Is your obtained r statistically significant (using a =.05, two tailed)? Are the Rater A and Rater B scores "reliable"? Is there good or…
6.2.8. Let X be N(0, 0), 0 <0 < x. (a) Find the Fisher information I(0). (b) If X₁, X2,..., Xn is a random sample from this distribution, show that the mle ofis an efficient estimator of 0. (c) What is the asymptotic distribution of √(-0)?

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9.6 Suppose that Y₁, Y₂,... Y, denote a random sample of size n from a Poisson distribution (Y₁ + Y₂)/2 and 2₂ = Y. Derive the efficiency of 2₁ relative with mean λ. Consider ₁ = to Â₂.