I have $N$ random variables $X_1, \dots, X_N$ that are independent and identically distributed. Define the quantity $y_i = \mathbb{E}\left[ \frac{X_i}{\sum_{j=1}^N \alpha_j X_j}\right]$, where $\mathbb{E}[\cdot]$ is the expected value and $\{\alpha_j\}_{j=1}^N$ are real numbers such that $\sum_{j=1}^N \alpha_j = 1$.
Can we say that all $y_i$ are equal to each other? I'm looking for a proof or counter-example.
On
Finally, $y_1$ and $y_2$ are not identically distributed in general. Take $N=2$ and $X_i\sim N(0,1).$ Therefore $X_1/X_2=Y$ is standard Cauchy distributed. But with $\alpha_1=1/3$ and $\alpha_2=2/3$ $$y_1=\frac{3}{1+2/Y},\ \ y_2=\frac{3}{Y+2}.$$ Saying that $y_1\sim y_2$ is saying that $1+\frac{2}{Y}\sim 2+Y$ or $\frac{2}{Y}\sim 1+Y$ which is not true since their respective characteristic functions are $e^{-2|t|}$ and $e^{it-|t|}.$
I apologize since I missread the question, calling $y_i=\frac{3X_i}{X_1+2X_2}$ what was rather their expectation. An extra difficulty is that the example provided for showing that $y_1$ and $y_2$ as above have different laws is the fact that they have no expectation. Therefore for really answering to the question by providing a counter example we take $\alpha_1 =1/3$ and $\alpha_2=2/3$ again but $X_1$ and $X_2$ independent and uniform on $[0,1]$, obtaining
$$E(\frac{3X_1}{X_1+2X_2})-E(\frac{3X_2}{X_1+2X_2})=3\int_0^1\int_0^1\frac{x_1-x_2}{x_1+2x_2}dx_1dx_2=0.1611...>0.$$