Let X = (X1, X2, ..., Xn)T be a random vector with mean vector μ and covariance matrix Σ. Suppose that for every a = (a1, a2, ..., an)T ∈ Rn, the random variable aTX has a (one-dimensional) Gaussian distribution on R. a) For fixed a ∈ Rn, compute the moment generating function MaTX(u) of the random variable aTX, writing the answer using μ & Σ. b) Define MX(t1, t2, ... , tn), the moment generating function of the random vector X, and express it in terms of MtTX, the moment generating function of tTX 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.
Let X = (X1, X2, ..., Xn)T be a random
a) For fixed a ∈ Rn, compute the moment generating
b) Define MX(t1, t2, ... , tn), the moment generating function of the random vector X, and express it in terms of MtTX, the moment generating function of tTX 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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