Discussion
The characteristic function of a vector \(\underline{X} \sim N_{p}(\underline{\mu}, \Sigma)\) is given by :
Answer: B
The characteristic function is \(e^{i\mu' t - \frac{1}{2}t'\Sigma t}\).
For \(X \sim N_{p}(\mu, \Sigma)\), \(t'X\) is univariate normal with mean \(t'\mu\) and variance \(t'\Sigma t\). The characteristic function of \(N(m, v)\) at 1 is \(e^{im - v/2}\), so
\[ E[e^{it'X}] = e^{i\mu' t - \frac{1}{2}t'\Sigma t} \]
Option A is the moment generating function evaluated wrongly (the m.g.f. is \(e^{\mu' t + \frac{1}{2}t'\Sigma t}\)). Options C and D put \(i\) on the quadratic term, which would make the characteristic function have modulus 1 everywhere, impossible for a non-degenerate normal.
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