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For every $\lambda \in {\mathbb{R}}^2$, the scalar ${\lambda }^{T}X$ is normal with mean ${\lambda }^T\mu$ and variance ${\lambda }^{T}V\lambda$. The moment-generating function of a normal distribution therefore givesThus $\mu$ and $V$ determine the moment-generating function on all of ${\mathbb{R}}^2$. By the uniqueness theorem for moment-generating functions, they uniquely determine the probability distribution of $X$.

Any unbiased linear estimator has the form considered in part (b), and its error is the linear combination of independent normal random variablesThe moment-generating function of a normal distribution therefore gives, for every nonzero real $\theta$,Since the exponential function is strictly increasing and ${\theta }^2{\sigma }^2/2>0$, minimizing this exponential moment is exactly the same as minimizing ${\sum }_i{a}_i^2$. Part (b) shows that the unique minimizer is $a_i=x_i/S_{xx}$. Thus, independently of the sign or magnitude of $\theta$,