Question 2. Consider a p-dimensional response variable j, containing p variables, with n observation vectors j1,..., In. The sample mean vector of these observation vectors is denoted by j = (1/n) -1 Ii- Consider a linear combination v of the observation vector j, defined as v = a1y1 + a2y2 +.+ apYp = ā' ÿ where aT = (a1, a2, . . . , ap). i) Show that the mean of v is ī = a 1 ü) Using the results of the previous question and given that E, (vi – 0)? n - 1 show that s? = aTSā where S is the sample covariance matrix of the random vector j.
Question 2. Consider a p-dimensional response variable j, containing p variables, with n observation vectors j1,..., In. The sample mean vector of these observation vectors is denoted by j = (1/n) -1 Ii- Consider a linear combination v of the observation vector j, defined as v = a1y1 + a2y2 +.+ apYp = ā' ÿ where aT = (a1, a2, . . . , ap). i) Show that the mean of v is ī = a 1 ü) Using the results of the previous question and given that E, (vi – 0)? n - 1 show that s? = aTSā where S is the sample covariance matrix of the random vector j.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please Solve this question correctly in one hour please make sure solve correctly
![Question 2. Consider a p-dimensional response variable j, containing p variables, with n observation
vectors j1,... , jn. The sample mean vector of these observation vectors is denoted by j = (1/n) E ỹ.
Consider a linear combination v of the observation vector j, defined as
v = a1y1 + a2Y2 + · ··+ ap Yp =
where aT = (a1, a2, . .. , ap).
i) Show that the mean of v is ī = aTj
ii) Using the results of the previous question and given that
E, (vi – 0)²
-
п — 1
show that
s? = aT Sā
where S is the sample covariance matrix of the random vector j.
[Hint: Try writing all the v variables in terms of the corresponding vectors j.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff8268f00-1e22-49fe-9b99-9cbbb3c8020a%2F22e796fb-d503-42bd-91b6-bbb8ea4fa593%2Flgo7tz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 2. Consider a p-dimensional response variable j, containing p variables, with n observation
vectors j1,... , jn. The sample mean vector of these observation vectors is denoted by j = (1/n) E ỹ.
Consider a linear combination v of the observation vector j, defined as
v = a1y1 + a2Y2 + · ··+ ap Yp =
where aT = (a1, a2, . .. , ap).
i) Show that the mean of v is ī = aTj
ii) Using the results of the previous question and given that
E, (vi – 0)²
-
п — 1
show that
s? = aT Sā
where S is the sample covariance matrix of the random vector j.
[Hint: Try writing all the v variables in terms of the corresponding vectors j.]
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