Homework Help is Here – Start Your Trial Now!
Math
Statistics
Q 6.3. Let X (X1, X2,..., Xn) be a random vector with mean vector and variance- (a1, a2,..., an)TER", the random variable covariance matrix E. Suppose that for every a = a X has a (one-dimensional) Gaussian distribution on R. (a) For fixed a ER", compute the moment generating function Marx (u) of the random variable a X writing the answer using and E. (b) Define Mx (t₁, t2, ..., tn), the moment generating function of the random vector X, and ex- press it in terms of Mr x the moment generating function of t X where t = (t1, t2,..., tn). (c) Combining (a) and (b), compute the moment generating function Mx (t) of X and hence prove that the random vector X has a Gaussian distribution.

Q 6.3. Let X (X1, X2,..., Xn) be a random vector with mean vector and variance- (a1, a2,..., an)TER", the random variable covariance matrix E. Suppose that for every a = a X has a (one-dimensional) Gaussian distribution on R. (a) For fixed a ER", compute the moment generating function Marx (u) of the random variable a X writing the answer using and E. (b) Define Mx (t₁, t2, ..., tn), the moment generating function of the random vector X, and ex- press it in terms of Mr x the moment generating function of t X where t = (t1, t2,..., tn). (c) Combining (a) and (b), compute the moment generating function Mx (t) of X and hence prove that the random vector X has a Gaussian distribution.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
See similar textbooks
icon
Related questions
Question

Q6.3

Q 6.3. Let X (X₁, X2,..., Xn)T be a random vector with mean vector μ and variance- covariance matrix E. Suppose that for every a = (a₁, A2,..., , an)¹ € R¹, the random variable T a¹ X has a (one-dimensional) Gaussian distribution on R. = (a) For fixed a € R", compute the moment generating function Marx (u) of the random variable a X writing the answer using µ and E. aT X (b) Define Mx (t₁, t2, ..., tn), the moment generating function of the random vector X, and ex- press it in terms of Mtrx the moment generating function of t X where t = (t₁, t2,..., tn). T (c) Combining (a) and (b), compute the moment generating function Mx(t) of X and hence prove that the random vector X has a Gaussian distribution.
Transcribed Image Text:Q 6.3. Let X (X₁, X2,..., Xn)T be a random vector with mean vector μ and variance- covariance matrix E. Suppose that for every a = (a₁, A2,..., , an)¹ € R¹, the random variable T a¹ X has a (one-dimensional) Gaussian distribution on R. = (a) For fixed a € R", compute the moment generating function Marx (u) of the random variable a X writing the answer using µ and E. aT X (b) Define Mx (t₁, t2, ..., tn), the moment generating function of the random vector X, and ex- press it in terms of Mtrx the moment generating function of t X where t = (t₁, t2,..., tn). T (c) Combining (a) and (b), compute the moment generating function Mx(t) of X and hence prove that the random vector X has a Gaussian distribution.
Expert Solution
steps

Step by step

Solved in 5 steps with 25 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman
SEE MORE TEXTBOOKS