  Due to the limited number of bits in floating-point or fixed-point
arithmetic, rounding is a necessary step in many computations. Although
rounding methods can be tailored for different applications, round-off errors
are generally unavoidable. When a sequence of computations is implemented,
round-off errors may be magnified or accumulated. The magnification of
round-off errors may cause serious failures. Stochastic rounding (SR) was
introduced as an unbiased rounding method, which is widely employed in, for
instance, the training of neural networks (NNs), showing a promising training
result even in low-precision computations. Although the employment of SR in
training NNs is consistently increasing, the error analysis of SR is still to
be improved. Additionally, the unbiased rounding results of SR are always
accompanied by large variances. In this study, some general properties of SR
are stated and proven. Furthermore, an upper bound of rounding variance is
introduced and validated. Two new probability distributions of SR are proposed
to study the trade-off between variance and bias, by solving a multiple
objective optimization problem. In the simulation study, the rounding variance,
bias, and relative errors of SR are studied for different operations, such as
summation, square root calculation through Newton iteration and inner product
computation, with specific rounding precision.
