1989 Putnam B6

Let $(x_1,\,x_2,\,\ldots\,x_n)$ be a point chosen at random from the

$n$-dimensional region defined by

$0<x_1<x_2<\cdots < x_n<1.$ Let $f$ be a continuous function on

$[0,1]$ with $f(1)=0$.

Set $x_0=0$ and $x_{n+1}=1$. Show that the expected value of the

Riemann sum

$$

\sum_{i=0}^n (x_{i+1}-x_i) f(x_{i+1})

$$

is $\int_0^1 f(t)P(t)\, dt$, where $P$ is a polynomial of degree $n$,

independent of $f$, with $0\le P(t)\le 1$ for $0\le t \le 1$.

令 $(x_1,\,x_2,\,\ldots\,x_n)$ 为从

$n$ 维区域定义为

$0<x_1<x_2<\cdots < x_n<1.$ 设 $f$ 为连续函数

$[0,1]$ 且 $f(1)=0$。

设置 $x_0=0$ 和 $x_{n+1}=1$。表明期望值

黎曼和

$$

\sum_{i=0}^n (x_{i+1}-x_i) f(x_{i+1})

$$

是 $\int_0^1 f(t)P(t)\, dt$，其中 $P$ 是 $n$ 次多项式，

独立于$f$，$0\le P(t)\le 1$ 对于$0\le t \le 1$。