Spatio-temporal development of electron swarms in gases is simulated using a propagator method based on a series of one-dimensional spatial moment equations. When the moments up to a sufficient order are calculated, the spatial distribution function of electrons, p( x), can be constructed by an expansion technique using Hermite polynomials and the weights of the Hermite components are represented in terms of the electron diffusion coefficients. It is found that the higher order Hermite components tend to zero with time; that is, the normalized form of p( x) tends to a Gaussian distribution. A time constant of the relaxation is obtained as the ratio of the second- and third-order diffusion coefficients, . The validity of an empirical approximation in time-of-flight experiments that treats p( x) as a Gaussian distribution is indicated theoretically. It is also found that the diffusion coefficient is defined as the derivative of a quantity called the cumulant which quantifies the degree of deviation of a statistical distribution from a Gaussian distribution.