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expected value of sum equals integral

Source: Putnam 1989 B6

August 27, 2021
probabilityexpected valuecalculusintegration

Problem Statement

Let (x1,x2,…,xn)(x_1,x_2,\ldots,x_n) be a point chosen at random in the nn-dimensional region defined by 0<x1<x2<…<xn<10<x_1<x_2<\ldots<x_n<1, denoting x0=0x_0=0 and xn+1=1x_{n+1}=1. Let ff be a continuous function on [0,1][0,1] with f(1)=0f(1)=0. Show that the expected value of the sum ∑i=0n(xi+1−xi)f(xi+1)\sum_{i=0}^n(x_{i+1}-x_i)f(x_{i+1})is ∫01f(t)P(t)dt\int^1_0f(t)P(t)dt., where PP is a polynomial of degree nn, independent of ff, with 0≤P(t)≤10\le P(t)\le1 for 0≤t≤10\le t\le1.
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