Common random numbers

Common random numbers couples simulations of different systems by reusing the same random inputs when estimating a difference. The individual output probability distributions must stay correct; the useful change is their dependence. If paired outputs are $X,Y$ with finite variances, thenPositive covariance therefore reduces the variance of their difference compared with independent outputs having those same marginal laws. This identity explains the technique, but does not guarantee a beneficial covariance for every simulation.^[ref]

For an original example, let $U$ have the uniform distribution on $[0,1]$, and compare $X=U$ with $Y=2U$. Their difference has expected value $-1/2$ and variance $1/12$. With independent uniforms $U,V$, the same marginal outputs $X=U$ and $Y=2V$ instead giveAveraging $n$ independent pairs divides either variance by $n$. In this stipulated example, sharing the input reduces variance fivefold for the same number of pairs.

The sign matters. If outputs are $U$ and $1-U$, sharing $U$ gives difference $2U-1$ with variance $1/3$. Using independent inputs gives variance $1/6$, so this coupling is worse. Reusing an input is useful when it produces an appropriate response in both systems, not simply because the same seed appears in two runs. In a simulation with branching or different event counts, matching random inputs to corresponding events is part of specifying the coupling.^[ref]