CMP.9:5.2 - Concentrate draws without changing an expectation

The target is the mean of values 0,0,0,4 under the uniform law on four identifiable outcomes, so μ=1. Direct draws have variance 3 per contribution.

Choose proposal probabilities q=(1/8,1/8,1/8,5/8), concentrating effort on the nonzero contribution. The corrected value on outcome 4 is 4*(1/4)/(5/8)=8/5; the other corrected values are zero. Its expectation is (5/8)*(8/5)=1, and its variance is (5/8)*(8/5)²-1=3/5. At the same number of independent draws, this construction reduces variance by a factor of five; the proposal and weighting cost decide the actual work benefit.

Using the unweighted sample mean instead has expectation (5/8)*4=5/2, the wrong target. Increasing the sample count would concentrate that wrong answer. If the required statistic changes to a different function on the four outcomes, retain the proposal only after recomputing its support, corrections and variation for that function.