Shows realizations, covariance functions, and spectral densities of two processes. One of the processes has continuous time and one has discrete time; both realizations have been smoothed to a...

Shows realizations, covariance functions, and spectral densities of two processes. One of the processes has continuous time and one has discrete time; both realizations have been smoothed to a continuous curve.

(a) For each plot, A–F, determine whether it depicts a realization, a covariance function, or a spectral density. Justify your answer!

(b) Which realization belongs to which covariance function and which spectral density?

(c) Suppose now that we would like to estimate the expected mean value of the processes with the average of only two samples at a time distance h. That is, we estimate
 Determine, for each process, an optimal value of h, i.e., the h giving the minimum variance of the estimator.