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
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
Q6.3

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

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step 1: Write the given information
VIEWStep 2: Compute the moment generating function Ma^TX(u) of the random variable a^TX
VIEWStep 3: Define Mx(t1, t2, ..., tn), the moment generating function of the random vector X
VIEWStep 4: Compute the moment generating function MX(t) of X combining (a) and (b)
VIEWSolution
VIEWStep by step
Solved in 5 steps with 25 images

Recommended textbooks for you

MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc

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
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning

MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc

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
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning

Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON

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
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman