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 variableTa¹ X has a (one-dimensional) Gaussian distribution on R.=(a) For fixed a € R", compute the moment generating function Marx (u) of the randomvariable 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 henceprove that the random vector X has a Gaussian distribution.

Given that be a random vector with mean vector and variance-covariance matrix .

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.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 32E

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