TITLE: Expectation of Minimum of $n$ i.i.d. uniform random variables.
QUESTION [54 upvotes]: $X_1, X_2, \ldots, X_n$ are $n$ i.i.d. uniform random variables. Let $Y = \min(X_1, X_2,\ldots, X_n)$. Then, what's the expectation of $Y$(i.e., $E(Y)$)?
I have conducted some simulations by Matlab, and the results show that $E(Y)$ may equal to $\frac{1}{n+1}$. Can anyone give a rigorous proof or some hints? Thanks!

REPLY [10 votes]: An short proof without the use of the pdf. Suppose $X_i \sim U(0,1)$. Then
\begin{align*}
E[\min(X_i)] &= \int_0^1 P[\min(X_i)>\mu] d\mu \\ 
&= \int_0^1 \prod_i P[X_i>\mu] d\mu \\ 
&= \int_0^1 (1-\mu)^n d\mu\\
&= \int_0^1 \mu^n d\mu\\
&= \dfrac{1}{n+1}
\end{align*}