  Most evolutionary algorithms have parameters, which allow a great flexibility
in controlling their behavior and adapting them to new problems. To achieve the
best performance, it is often needed to control some of the parameters during
optimization, which gave rise to various parameter control methods. In recent
works, however, similar advantages have been shown, and even proven, for
sampling parameter values from certain, often heavy-tailed, fixed
distributions. This produced a family of algorithms currently known as "fast
evolution strategies" and "fast genetic algorithms".
  However, only little is known so far about the influence of these
distributions on the performance of evolutionary algorithms, and about the
relationships between (dynamic) parameter control and (static) parameter
sampling. We contribute to the body of knowledge by presenting, for the first
time, an algorithm that computes the optimal static distributions, which
describe the mutation operator used in the well-known simple $(1+\lambda)$
evolutionary algorithm on a classic benchmark problem OneMax. We show that, for
large enough population sizes, such optimal distributions may be surprisingly
complicated and counter-intuitive. We investigate certain properties of these
distributions, and also evaluate the performance regrets of the $(1+\lambda)$
evolutionary algorithm using commonly used mutation distributions.
