. 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)
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Chapter1: Combinatorial Analysis
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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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