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

December 14, 2015 · by randomprojection · in Probability · Leave a comment

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 $E[Z_n]$ .