谈谈stratified sampling 蒙特卡洛数值积分方法，给我一个例子

Stratified sampling is a Monte Carlo numerical integration technique used to reduce variance in the estimation of the integral of a function. The basic idea is to divide the domain of integration into several non-overlapping subdomains, or strata, and then sample from each subdomain separately.

For example, suppose we want to estimate the integral of the function f(x) = x^2 over the interval [0,1]. We can divide the interval into two equal subintervals [0,0.5] and [0.5,1]. We then take a random sample of size n/2 from each subinterval, where n is the total sample size we want to use.

The estimator of the integral is then the average of the sample means of each subinterval, weighted by the size of each subinterval. That is,

∫[0,1] f(x) dx ≈ (0.5 ∫[0,0.5] f(x) dx + 0.5 ∫[0.5,1] f(x) dx)

≈ (0.5/n)Σf(xi) + (0.5/n)Σf(xi)

where xi denotes the ith sample from the corresponding subinterval.

Stratified sampling can reduce variance compared to simple random sampling if the function has large variations over the domain of integration. It ensures that each subinterval is represented in the sample, and thus reduces the risk of under-sampling regions of high variations.