The stochastic evolution in space of an electron avalanche has been investigated with the use of Monte Carlo methods. From these investigations we have obtained the energy distribution as a function of distance from the cathode and avalanche size, f(n,ε,x) the probability distribution for avalanche size, P(n,x); and the ionization probability, per unit distance at x, for electrons belonging to an avalanche of size n, alphas(n,x). We believe that this is the first time that any information has been presented on f(n,ε,x) and alphas(n,x) which takes microscopic processes into account. The investigations were done in nitrogen for E/N>=300 Td. We suggest that pulse-height distribution experiments may be a sensitive technique for determining the E/N conditions at which runaway electrons appear.