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n 个 U(0,1) 的最小值期望

What is the expected value of their minimum?

专题
Probability / 概率
难度
L4

题目详情

Assume that X1,X2,…,XnX_{1},X_{2},\ldots ,X_{n} are independent uniform random variables on [0, 1], where n≥2n\geq 2 . What is the expected value of their minimum?

解析

设 M=min(X1,ldots,Xn)M=\\min(X_1,\\ldots,X_n)。则\n\n\nmathbbP(M>t)=(1−t)n,quad0letle1.\n\n\\mathbb{P}(M>t)=(1-t)^n,\\quad 0\\le t\\le 1.\n\n\n因此\n\n\nboxedmathbbE[M]=int01(1−t)ndt=frac1n+1.\n\n\\boxed{\\mathbb{E}[M]=\\int_0^1(1-t)^n dt=\\frac{1}{n+1}}.\n