Let $X_{1},X_{2}$ and $X_{3}$ are three $U(0,1)$ random variable, then find the $E\left[\frac{X_{1}+X{2}}{X_{1}+X_{2}+X_{3}}\right]$.
My question is can I distribute the expectation in the numerator and denominator. But, I am not able to proceed from there.
How the find the distribution of ratio of sum of uniform Variable so that we can find its expectation. Please help. I am really stuck at this.
Thanks
You may not distribute the expectation into the denominator. Don't ever do that.
Define $Z_i =\frac {X_i}{X_1+X_2+X_3}.$ So you want to find $E(Z_1+Z_2).$ Here's the trick: we have $Z_1+Z_2+Z_3=1,$ so $$ E(Z_1)+E(Z_2)+E(Z_3)=1$$ (since we can distribute an expectation over a sum... linearity of expectation)
Now we get to the point where I ask you "are you sure you aren't allowed to make any assumptions about the $X_i$ that you forgot to tell me about, since we can't finish the problem without making some?" and then you say "oh yeah, they are independent."
And then we see (think symmetry) that this implies $Z_1,$ $Z_2$ and $Z_3$ are identically distributed. And then the answer is easy to see from the equation above (think symmetry again).
(And actually, you don't need the $X_i$ to be independent: the weaker condition of exchangeability suffices.)