Suppose I have $a_1,...a_n \in [0,1]$. From there I chose $m$ numbers randomly, independent of each other. What is the expected value of the sum?
My approach: let $X_i =1$ if $a_i$ is chosen. Then $\sum_{n=1}^{n} X_i = m$. Then $S = \sum_{n=1}^{n} a_i X_i$ and $E[S] = \sum_{n=1}^{n} a_i E[X_i]$. How to calculate $E[X_i]$, subject to the contraint the $m$ numbers are chosen?