optimize/ae p39 beat the average

optimize/ae p39 beat the average: a task in MathConstructOptimize-Envs-harbor (Harbor dataset). Let X1, X2, X3, X4 be independent copies of a random variable X taking non-negative integer values. Find the distribution of X that maximizes

The task

Let X1, X2, X3, X4 be independent copies of a random variable X taking non-negative integer values. Find the distribution of X that maximizes

Part of amphora/MathConstructOptimize-Envs-harbor.