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. Assume that {(X;}", for integer m, are a random sample from a Gamma(n, 1/k) distribution, where n is also an integer and > 0. Determine the large sample approximation to the joint distribution of Vn{X;/n -k} noting any first and second moments explicitly in this distribution. Define a new vector V whose entries are Y, = E X{ for j = 1,..,4. Determine the mean and covariance of V. You may use results on the 1st and 2nd moments of the Gamma distribution from the course formula sheet on moodle without re-deriving the results, and more generally that the Ith uncentered moment of the Gamma(a, 8) distribution takes the form (for lE N):

. Assume that {(X;}", for integer m, are a random sample from a Gamma(n, 1/k) distribution, where n is also an integer and > 0. Determine the large sample approximation to the joint distribution of Vn{X;/n -k} noting any first and second moments explicitly in this distribution. Define a new vector V whose entries are Y, = E X{ for j = 1,..,4. Determine the mean and covariance of V. You may use results on the 1st and 2nd moments of the Gamma distribution from the course formula sheet on moodle without re-deriving the results, and more generally that the Ith uncentered moment of the Gamma(a, 8) distribution takes the form (for lE N):

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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5. Assume that {X;}", for integer m, are a random sample from a Gamma(n, 1/k) distribution, where n is also an integer and x > 0. Determine the large sample approximation to the joint distribution of Vn{X;/n - k} noting any first and second moments explicitly in this distribution. Define a new vector V whose entries are Y; = E, X for j = 1,..., 4. Determine the mean and covariance of V. You may use results on the 1st and 2nd moments of the Gamma distribution from the course formula sheet on moodle without re-deriving the results, and more generally that the Ith uncentered moment of the Gamma(a, 8) distribution takes the form (for lE N): E(x'} = | -Bs dr e r(a) r(a +1) T(a)) J. r(a+ 1) r(a +1) r(a)3 lta-1 e dr 1= 1,2, 3, ... r(a+l) We recognize Ta = (a +1- 1)a-1 as the falling factorial for integer a. %3D
Transcribed Image Text:5. Assume that {X;}", for integer m, are a random sample from a Gamma(n, 1/k) distribution, where n is also an integer and x > 0. Determine the large sample approximation to the joint distribution of Vn{X;/n - k} noting any first and second moments explicitly in this distribution. Define a new vector V whose entries are Y; = E, X for j = 1,..., 4. Determine the mean and covariance of V. You may use results on the 1st and 2nd moments of the Gamma distribution from the course formula sheet on moodle without re-deriving the results, and more generally that the Ith uncentered moment of the Gamma(a, 8) distribution takes the form (for lE N): E(x'} = | -Bs dr e r(a) r(a +1) T(a)) J. r(a+ 1) r(a +1) r(a)3 lta-1 e dr 1= 1,2, 3, ... r(a+l) We recognize Ta = (a +1- 1)a-1 as the falling factorial for integer a. %3D
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