In this paper, We show that the bandpass random signals of the form ∑$_{alpha}$$alpha$$_{alpha}$ a Sin(2$pi$f$_{alpha}$t + b$_{alpha}$) where a$_{alpha}$ being a random number in [0,1], f$_{alpha}$ a random integer in a given frequency band, and b$_{alpha}$ a random number in [0, 2$pi$], generate Gaussian white noise signals and hence they are adequate for simulating Continuous Markov processes. We apply the result to the fluctuation-dissipation formula for the Johnson noise and show that the probability distribution for the long term average of the power of the Johnson noise is a X$^2$ distribution and that the relative error of the long term average is (equation omitted) where N is the number of blocks used in the average.error of the long term average is (equation omitted) where N is the number of blocks used in the average.