Reference functions to decrease errors in Monte Carlo integrals

Example of Monte Carlo integrals where reference functions are useful

“…Since the functions to be integrated in this approximation are easily integrable polynomials of cos 0*, they give us a hint that powers of cos 0* may have the properties recommended in Ref. [1], Sect. 2.4, for efficient reference functions.…”

“…As stated in Ref. [1], the same statement should apply whenever computations involve integrating functions that can be approximated, at most of the Monte Carlo points, by a linear combination of functions whose integral can be expressed in closed form, i.e., of reference functions. Distributions of the quantity x;in computed with use of reference functions from Monte Carlo events produced either by a) an unbiased generator orb) a biased generator.…”

“…[1], Sect. 3.1, it was stated that reference functions should make the result less sensitive to some systematic errors in the Monte Carlo generation.…”

“…[1], a clear distinction is made betwP.en the random variables such as eCMS and the non-random quantities such as the values XCMS that the random variables (.CMS can take. Here, it was feared that keeping up with this convention entirely may obscure the physical content of the example.…”