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An analysis of digital random numbers generated from serial samples of Gaussian noise is presented. A methodology is developed for calculating the word probability distribution in terms of the autocorrelation function of the noise. Examples are presented for three and four bit words. A related problem is also considered which concerns the noise autocorrelalion function. Bounds are derived which take into account nonideal effects of the filter used to shape the noise spectrum. An example is given for a modified double-pole bandpass model.