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.

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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.
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