Expected value : Reciprocal Sum Positive iid Random Variables Let $\{X_i\}_{i=1}^{n}$ be iid positive valued random variables. Show that $\mathbb{E}[\frac{\sum_{i=1}^{m}X_i}{\sum_{i=1}^{n}X_i}]=\frac{m}{n}$ for $1\leq m\leq n$. 
I know $\mathbb{E}[\frac{\sum_{i=1}^{m}X_i}{\sum_{i=1}^{n}X_i}]=m \mathbb{E}[\frac{X_i}{\sum_{i=1}^{n}X_i}]$. 

If you can prove that $\mathbb{E}[\frac{\sum_{i=1}^{m}X_i}{\sum_{i=1}^{n}X_i}]=m \mathbb{E}[\frac{X_i}{\sum_{i=1}^{n}X_i}]$, then simply find $$1=\mathbb{E}[\frac{\sum_{i=1}^{n}X_i}{\sum_{i=1}^{n}X_i}]=n \mathbb{E}[\frac{X_i}{\sum_{i=1}^{n}X_i}]$$ and conclude. 
