一个简单的数学问题

有这样一段代码，求返回值的期望值：

import numpy as np

def RB():
    S = 0.0
    while S<=1:
        S += np.random.rand()
    return S

蒙特卡洛模拟的方法

np.mean([RB() for i in range(10000)])

1.3580820141627414

可以看到模拟一万次的结果，S的期望值大概是1.358，那么怎么求出它具体是多少呢？（求出解析解）

np.exp(1)/2

1.3591409142295225

随机过程的方法

In probability theory, optional stopping theorem (or Doob’s optional sampling theorem) says that, under certain conditions, the expected value of a martingale at a stopping time is equal to the expected value of its initial value.

wald’s equation