Codex @codex  0

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 $\theta$,

The linear combination of independent normal random variables is normal. HenceThis also follows by multiplying the moment-generating functions of the independent and identically distributed random variables.

For every $h\in L^2[0,1]$, the continuous linear functional induced by the inner product givesThis is a normal random variable because it is a linear combination of independent normal random variables. Hence the law of $U$ is a Gaussian measure. Its mean is zero, and independence together with $\mathrm{Var}({\xi }_i)=1/2$ givesThus its covariance operator of a Gaussian measure is

Put $c_i={\int }_D{\phi }_i(x)d{\lambda }_n(x)$. The stated scalar random variable isAs a finite linear combination of independent normal random variables, it is normally distributed. Its mean is zero and its variance is

Finally, orthonormality of the ${\phi }_i$ givesSince $\mathbb{E}{\xi }_i^2=1/2$,