  The probability-generating function of the number of electron-positron pairs
produced in a uniform electric field is constructed. The mean and variance of
the numbers of pairs are calculated, and analytical expressions for the
probability of low numbers of electron-positron pairs are given. A recursive
formula is derived for evaluating the probability of any number of pairs. In
electric fields of supercritical strength |eE| > \pi m^2/ \ln 2, where e is the
electron charge, E is the electric field, and m is the electron mass, a
branch-point singularity of the probability-generating function penetrates the
unit circle |z| = 1, which leads to the asymptotic divergence of the cumulative
probability. This divergence indicates a failure of the continuum limit
approximation. In the continuum limit and for any field strength, the positive
definiteness of the probability is violated in the tail of the distribution.
Analyticity, convergence, and positive definiteness are restored upon the
summation over discrete levels of electrons in the normalization volume.
Numerical examples illustrating the field strength dependence of the asymptotic
behavior of the probability distribution are presented.
