How to calculate the expected value of $E\big(x_1/(x_1+x_2+x_3+\cdots+x_n)\big)\,$?

Question

I am struggling with this question.

Assume that $x_1, \cdots, x_n$ are iid variables from the uniform distribution $U(a,b)$, where $0<a<b$. How to calculate $E(x_1/\sum_{i=1}^{n} x_i)\,$?

I have seen here that it seems the answer is $1/n$, but i have no idea how to calculate it.

Answer 1

By the linearity of the mean, $$1 = E(1) = E \left( \dfrac{\sum_{i=1}^n x_i}{\sum_{i=1}^n x_i} \right) = \sum_{j=1}^n E\left( \dfrac{x_j}{\sum_{i=1}^n x_i} \right) = n \cdot E \left( \dfrac{x_1}{\sum_{i=1}^n x_i} \right),$$ where the last equality holds because the variables are identically distributed. Finally, solving for $E \left( \dfrac{x_1}{\sum_{i=1}^n x_i} \right)$, we obtain $E \left( \dfrac{x_1}{\sum_{i=1}^n x_i} \right) = \dfrac{1}{n}$.

Answer 2

Hint: Note that by symmetry, $\frac{x_1}{\sum_{i=1}^n x_i}$ has the same distribution as $\frac{x_2}{\sum_{i=1}^n x_i}, \frac{x_3}{\sum_{i=1}^n x_i}, \ldots$, so they must all also have the same expectation. What is their sum, and how can that help you compute each individual expectation?