Expectation of minimum of n i.i.d. uniform random variables

Let X_1,\ldots,X_n be independent uniform random variables from [0,1], and consider the random variable Z_n=\min (X_1,\ldots,X_n). Computing the expectation E[Z_n] is a routine computation: E[Z_n]=\int_0^1 Pr(X>x)^n dx=\int_0^1 (1-x)^n dx = \frac{1}{n+1}.

However, there a slick way of computing this expectation. Let X_{n+1} be another uniform random variable in [0,1]. Consider the probability Pr(X_{n+1}<Z_n). On the one hand due to symmetry, it is equal to \frac{1}{n+1}, on the other hand it is also equal to E[Z_n].

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