For the case of f(x)/p2(x), in the left region where p2(x) is very small and f(x) is large, the ratio will end up being a very large number. This means that samples taken in this region will have a very large variance in their result and will greatly impact on the Monte Carlo estimate especially if there are any errors in p1(x) or f(x) measurements.

adityasingh7311

The peaks and valleys of p1 more closely correspond to the peaks and valleys of f whereas p2 is almost inversely related and is high where f has the lowest values and low where f has the highest values. This makes p2 worse than p1 since p2 is more likely to have us sample the low points of f that don't contribute much area and less likely to correspond to the higher points of f that do contribute to the overall area.

For the case of f(x)/p2(x), in the left region where p2(x) is very small and f(x) is large, the ratio will end up being a very large number. This means that samples taken in this region will have a very large variance in their result and will greatly impact on the Monte Carlo estimate especially if there are any errors in p1(x) or f(x) measurements.

The peaks and valleys of p1 more closely correspond to the peaks and valleys of f whereas p2 is almost inversely related and is high where f has the lowest values and low where f has the highest values. This makes p2 worse than p1 since p2 is more likely to have us sample the low points of f that don't contribute much area and less likely to correspond to the higher points of f that do contribute to the overall area.