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This paper considers the problem of interpolation and extrapolation of a band-limited random process from a finite set of N equally spaced samples using a linear mean-square estimation approach. Various theoretical properties of the solutions are discussed, such as the effect of sample spacing, number of samples, and bandwidth. Computer solutions are presented that are useful for both illustrating the theoretical behavior and providing quantitative measures of the interpolation and extrapolation error. It is demonstrated that Nyquist sampling at twice the highest frequency is not an appropriate criterion for finite N: undersampling can give satisfactory performance for a band-pass process, while oversampling can be required for a low-pass process. These conclusions are similar to known results for sampling over infinite intervals.