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n 个 U(0,1) 的最大值期望

What is the expected value of their maximum

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 maximum?

解析

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